HUME LIBRARY
INSTITUTE OF FOOD AND
AGRICULTURAL SCIENCES
UNIVERSITY OF FLORIDA
Gainesville
Digitized by the Internet Archive
in 2013
http://archive.org/details/fieldplottechniqOOIeon
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4
FIELD PLOT TECHNIQUE
by
WARREN H. LEONARD, M. Sc. (Nebraska)
Professor of Agronomy
Colorado State College
and
ANDREW G. CLARK, M. A. (Colorado)
Professor of Mathematics
Colorado State College
Copyright 1939
by
Warren H. Leonard
and
Andrew G. Clark
1946 Printing
BURCESS PUBLISHING CO.
426 SOUTH SIXTH STREET
MINNEAPOLIS 15, MINN.
•
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PREFACE
This manual has "been the outgrowth of a set of lectures on Field Plot Technique
given to seniors and graduate students at Colorado State College since 1930* It has
heen found practical in the classroom for a 2 to k credit combined lecture and labor
atory course. The problems and questions have proved to be important aids to the
student. While "Field Plot Technique" has been prepared primarily for clas3 use, it
is hoped that it will appeal to the technical worker in Agronomy as a reference to
the more important statistical methods and tables. The large number of references
quoted will give the reader a ready reference to the major papers on various phases
of applied statistics.
The organization of the subject matter, and the manner in which the statistical
methods are interwoven with the applications, differs somewhat from the conventional
approach. The writers feel that the student of agronomic experimentation needs an
elementary picture of the factors to be considered in a research program with special
reference to the field experiment. For this reason, an attempt has been made to
coordinate the historical and logical background of agronomic experimentation with
statistical techniques and their application to the design of the practical types of
field experiments. This also requires that the student be familiar with the mechan
ical procedures generally followed in routine experimental work.
The development of the various statistical techniques has been intuitive rather
than rigorously mathematical. The aim has been to lead the student to understand
the formulas he applies without necessarily being able to derive them mathematically.
The symbolism employed in the text was chosen with regard to what appears to be the
most common usage. Considerable effort has been spent in striving for consistency.
That it is impossible in an elementary text to present and interpret many of
the complexities involved in some modern experiments is obvious. It is hoped that
a sufficient foundation will be laid for the student so that he can intelligently
study the more advanced treatises.
The writers are deeply indebted to Dr. F. R. Immer, Professor of Agronomy and
Plant Genetics, University of Minnesota, for permission to make liberal use of his
classroom material, especially in chapters 11, 17, and 18. They wish to express
their appreciation to Dr. S. C. Salmon, Division of Cereal Crops and Diseases, U. S.
Department of Agriculture, for criticisms and helpful suggestions. Dr. K. S. Quisen
berry of the same division has assisted by his criticisms of chapter 21. The wri
' ters are particularly grateful to Professor R. A. Fisher and his publishers, Oliver
and Boyd, for permission to reproduce the Table ofx 2 from "Statistical Methods for
Research Workers." Professor G. W. Snedecor, Iowa State College, generously allowed
us to include his table of "F and t". The writers also wish to express their thanks
to Dr. C. I. Bliss for permission to use his table of angular transformations. The
table of Weparian logarithms used in the manual is taken from "Four Figure Mathemati
cal Tables" by the late J. T. Bottomley and published by Macmillan and Co., Ltd.
(London) . The writers are grateful to the publishers and to the representatives of
the author for permission to use this table. To Dr. D. W. Robertson, one of their
colleagues, they express their appreciation for various helpful suggestions.
S)
TABLE OF CONTENTS
Part I. Introduction to Experimentation
CHAPTER
I Status of Agronomic Research . ...... _ ... 1
II History of Basic Plant Sciences 9
III Logic in Experimentation _ 17
IV Errors in Experimental Work 28
Part II. S tatistical Analysis of Data
V Frequency Distributions and their Application 37
VI Tests of Si gni f 1 nance 5 k
VII The Binomial Distribution and its Applications 70
tJFHl The X 2 Test of Goodness of Fit and Independence. 75
IX Simple Linear Correlation 87
 X The Analysis of Variance 103
XI Covariance with Special Reference to Regression 113
Part III. F ield and Other Agronomic Experime nts
XII Soil Heterogeneity and its Measurement 131
XIII Size, Shape, and Nature of Field Plots lAl
XIV Competition and other Plant Errors 135
• XV Design of Simple Field Experiments I67
XVI Quadrat and other Sampling Methods 186
XVII The Complex Experiment  195
XVIII The Split Plot Experiment . 211
XIX Confounding in Factorial Experiments „ 221
XX Symmetrical Incomplete Block Experiments 250
XXI Mechanical Procedure in Field Experiments , 238
Part IV. Appendix
TABLE
1 Normal Probability Integral Table 251
2 Table of "F" and "t" 25^
3 TheX2 Table 258
Table of One half Naperian Logarithms 259
5 Table of the Angular Transformation 2o2
6 Table of Random Numbers 266
7 Index
ii
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FIELD PLOT TECHNIQUE
Part I
Introduction to Experimentation
CHAPTER I
STATUS OF AGRONOMIC RESEARCH
I. Rise of Agronomic Research
Although the art of agronomy has "been practiced for centuries, the science of agrono
my is only about 100 years old. The need for reliable information in this country
has come about gradually as farmers have come to realize some of the many problems
which confront the agricultural industry, problems in soil fertility, the control of
diseases and pests, winter hardiness in crops, among many others. In addition to
the needs of the farmers themselves, the establishment of the Land Grant Colleges
under the Morrill Act in 1862 brought about an acute need for subject matter for the
agricultural colleges . It soon became very apparent that the problems in agriculture
were complex and that well trained men were needed to solve them. In general, it
may be said that agricultural research began with simple empirical tests, but has
gradually developed until it has now attained a scientific basis. In the short space
of 75 years, so much subject matter has been accumulated in the field of agriculture
that no one man could hope to be familiar with all of it. This led to specialization
within the field between 1°00 and 1910 in America. The branches recognized in most
agricultural colleges and experiment stations are: Agronomy, animal husbandry,
horticulture, entomology, forestry, home economics, and veterinary medicine or path
ology.
Agronomy as a science was developed from the old style variety trials, crop rotation
tests, and soil culture experiments, when field culture was an empirical art. Re
search workers and others interested in the science of crops and soils formed the
American Society of Agronomy in 1907. In regard to Agronomy, Carleton (1907) states:
"As a science it investigates anything and everything concerned with the field crop,
and this investigation is supposed to be made in a most thorough manner, just as
would be done in any other science". Thus, agronomy is the laboratory and workship
of many sciences: Agrostology, chemistry, botany, ecology, genetics, pathology,
physics, physiology, and others concerned with the problems of crops and soils.
Ball (1916) early observed that it has been necessary for the experimenter (in
agronomy) to turn from the gross aspect to minute detail in order to solve some of
Its problems. Empirical knowledge has been rapidly supplemented by fundamental in
formation as a result of organized research and the improvement in its technique.
II. Establishment of Experiment St ations
It is difficult to realize that the present large network of experiment stations in
this country and in other parts of the world has been established in the past IOC
years. In fact, the science of agriculture practically began with this movement.
(a) First Experiment Station
Jean Baptiste Boussingault established the first experiment station in 183^,
being the first man to undertake field experiments on a practical scale. He farmed
land at Bechelbronne, Alsace, where he carried on research of a high calibre. Bous
singault set out to investigate the source of nitrogen in plants, and systematically
weighed the crops and the manures applied for them. He analyzed both and prepared a
balancesheet. Furthermore, this investigator studied the effects on plants when
legumes were in the rotation. He concluded that plants obtained most of their nitro
gen from the soil.
(See Chapter 2.)
1
(°) Rothamst gd Expe rimental Stat i on
The Rothamsted Experimental Station was established "by John Bsnnot Lowes on
his farm in England in iQKl . Hall (1905), in his account , states that "Rothamsted
is now a household word wherever the science of agriculture is studied." Lawes
found that phosphates wore important fertilizers and discovered a method to make
phosphate fertilizer "by the application of sulfuric acid to phosphate rock. Formerly
hones were used as a sole source of phosphates. This significant discovery led to
experimentation on a large scale. The systematic field experiments, "begun in 18^5
and continued to this day, have dealt particularly with soil fertilizers and crop
rotation. These experiments long have teen models for carefully planned, experiments.
Lawes was aided by Dr. J. H. Gilbert, who commenced work at Rothamsted in 1.8^3 • The
two men worked together for 57 years. Recently, Dr. R. A. Fisher has brought about
modifications in the field experiments to make them amenable to statistical treatment.
( c ) American Expe riment Stations
Some of the early history of American experiment stations is given by True
(1937) and by Shepardson (1929). South Carolina went on record as favoring an ex
periment station in 1785, but the general movement for the establishment of experi
ment stations began about I87I because of the attention attracted by the experiments
of Lawes and Gilbert of England. In the meantime, the Morrill Act signed by Lincoln
in 1862, provided for the socalled landgrant colleges for the study of agriculture
and mechanic arts. California established the first experiment station in lQl~), and
I began field experiments on deep and shallow plowing for cereals. A station was
started in North Carolina in I877, after which many others followed. The Hatch Act,
passed by Congress in I087, was the start of the present experiment stations. Twelve
were in existence at that time. Increased funds were provided by the Adams Act in
1906, by the Purnell Act in 1925, and by the Bankhead denes Act, in 1255. The United
I States Department of Agriculture has been of rather recent origin, the Secretary be
coming a Cabinet member in I889. At present, the federal government controls funds
given to the states for experimental work. In general, the system has been satis
factory because it has proved, to be participation and coordination rather than con
trol.
Ill . Reasons for Publi c Support of Agr icultural Research
There has been some criticism on the use of public funds for agricultural research,
but their use has been justified on the grounds that the welfare of agriculture is
basic to the nation. In addition, it would be almost impossible to place agricultur
al research on a selfsupporting "basis because the results of research are so diffi
cult to control through patents or otherwise.
( a) Agric ultural Welf are
There" are 6,000,000 farmers and 360,000,000 to 365,000,000 acres of culti
vated land in this country, some of which ha3 been cultivated more than 300 years.
The virgin fertility, in many cases, has been exhausted. The experiences and needs
of these farmers are significant because the prosperity of the nation depends to a
large extent upon agriculture. The production of food and fiber is fundamental to
the public welfare, as research that leads to lower cost of production passes its
benefits on to the consumer. Haskell (1923) calls attention to the fact that, in
the case of crop losses due to diseases and other factors, the consumer ultimately
pays a higher price for his food. He pays for depleted soil fertility in the same
way. Thus, the state may actually gain more from the benefits of research than the
farmer himself.
(b) Limitations of Farm Experience s
It has been impossible for several reasons to collect scientific information
of much value from farm experiences, (l) Inadequate Farm Records : The results ob
tained by farmers are inaccurately and incompletely recorded from the experimental
viewpoint. Their experiences are generally limited to acres and yields such as
found in stories in the farm press. Farmers very often place undue emphasis on the
unusual. (2) Failure to Consider all Factors ; The essence of scientific progress
is to determine "why". Among the many variables in agriculture, variation in season
is exceedingly important and may overshadow all other factor's. The farmer is quite
likely to base his judgment and conclusions on the results of one or two year's per
formance. Thome (1909) states that many experiments which farmers attempt are
valueless or misleading "because of failure to observe some essential condition of
experimentation. (3) Inadequate Training ; As a rule, the farmer lacks the training
or experience necessary for the evaluation of experimental results. Hall (1905)
makes this statement : "Agricultural science involves some of the most complex and
difficult problems the world is ever likely to have to solve, and if it is to con
tinue to be of benefit to the farmer, investigations, so far as their actual conduct
goes, must quickly pass into regions where only the professional scientific man can
hope to follow them ...." (h) Inadequate Funds ; Farmers lack the funds, help, and
equipment necessary for experimental work. Experimentation is quite expensive since
practical considerations are necessarily put aside. An experiment must be conducted
with precision in order to obtain reliable results, rather than for financial return.
For instance, Hall (1905) tells that some F.othamsted fields have grown wheat for 60
years, year after year, on the same land. As the modern farmer seldom grows wheat
continuously, he looks upon this experiment as hopelessly impractical when it is
pointed out to him on field days. Nevertheless, this very test furnished the bulk
of the early proof that losses in yield would result from continuous wheat culture.
The aim of the Fothamsted test, as it continues, is to find out how the wheat plant
grows .
IV. Experiment Station Funds
Agricultural research in this country is publicly financed almost altogether. Feder
al and state agencies spent about 25 million dollars on agricultural research for
the year 192728. This total sum represented approximately 0.20 percent of the gross
income for agricultural products, a figure wholly within reason.
^a) The Hatch ActV
~~TTTe f IrKtrTederal subsidy for agricultural research was the Hatch Act, passed
in I887. It gave each state $15,000 per year, a wide latitude in the use of the
funds being permitted. The Act made it possible to conduct original experiments or
verify experiments along lines as follows: (1) physiology of plants and animals;
(2) Diseases of plants and animals with remedies for the same; (5) the chemical com
position of useful plants at different stages of growth; (k) rotation studies;
(5) testing the adaptation of new crops and trees; (6) analyses of soils and water;
(7) chemical composition of manures, natural and artificial, and, their effect on
crops; (8) test the adaptation and value of grasses and forage plants; (9) test the
composition and digestibility of different foods for domestic animals; (10) research
.on butter and cheese production; and (11) examination and . classification of soils.
None of the funds can be used for the purchase or rental of lands or expenses for
farm operations .
(b) The Adams Act
A similar amount of money was granted to the states by the Adams Act, passed
in 1906. The funds must be used for original researches or experiments that bear
directly on agriculture. Research of a fundamental nature is required under this
forid. Norjs .of the money can be applied to substations, or to the purchase or rental
of land .
(c) The Purnell Act
The Purnel Act passed in 1925 provided for additional funds which now amount
to $60,000 per year for each state. These funds must "be used on specific projects,
hut the requirements are lees exact than for the use of Adams funds . The Act pro
vides for investigations on the production, manufacture, preparation, use, distribu
tion, and marketing of agricultural products.
( &) The Bankhead Jones Act '
Certain difficulties in the use of experimental funds for broad general pro
jects led to the passage of the Bankhead Jones Act in 1935 which will, in five years
(19^0), provide $5,000,000 for research. Its provisions have "been described as
follows: "To conduct scientific, technical, economic, and other research into laws
and principles underlying basic problems of agriculture in its broadest aspects....".
It also authorizes research for the improvement of quality of agricultural commodi
ties and for the discovery of uses for farm products and byproducts. The U. S.
Department of Agriculture receives ko per cent of this fund, while 60 per cent is
allotted to the states on the basis of rural population. It is generally understood
that the funds must be used for new lines of work.
V . The Personal Equation in Research
As for agricultural research in general, successful agronomic research depends upon
the ability, permanency, and honesty of the workers. The personnel for investiga
tional work must be welltrained in the basic sciences as well as leaders in agricul
tural thought. Their outlook must be broad.
(a) Education for Investigatio nal Work
The amount of training necessary for research is great. The investigator
must be skilled in the art of agronomy and trained in the closely related sciences.
In fact, he should have an adequate educational background before research is even
attempted. A good foundation in English, physics, and chemistry are basic for all
research in agriculture. Biology adds the conception of organism, while mathematics
is the common instrument. Thorough training in all branches of botanical science is
desirable in agronomy. This includes taxonomy, anatomy, physiology, pathology, etc.
Other sciences that are useful are: Geology, bacteriology, genetics, and statistics.
Among the authorities who agree on this general type of background are Howard (1924),
Wheeler (19U), Ball (1916), Carleton (1907), and Richey (1937). A practical view
point is necessary, but this is largely the result of boyhood training and common
sense.
(b) Qualities in Successful Research Men
There is some question about the successful scientist necessarily being a
genius. The term should be qualified to include perseverance, common sense, and in
finite pains. Howard (1924) emphasized the qualities needed when he said: "Here the
man is everything; the system is nothing." (1) Imagination : Some imagination is
essential in the research worker but, of course, it must be scientific imagination.
(2) D i s cr imlnat i on : An investigator must have the power of discrimination, that is,
he must be able to recognize the essentials and nonessentials in research. He must
select the features which are most worthwhile. It is possible to record too much
data on a subject, and thus cloud the entire issue. (3) Accuracy : There is a great
need for accuracy in' experimental work. An investigator should record only those
notes whose reliability is well established. One should never take measurements so
fine that they imply false accuracy. The figures taken by an investigator should
give him confidence in his work, (k) Honesty in observatio n:. The investigator should
always accept observations without regard to their agreement with his own precon
ceived ideas. One should record only the things he sees. (5) Fairness : The re
search man should give due credit to others, and keep within his own field unless a
phase of his work calls for cooperation with others. (6) Enthusiasm: One should he
enthusiastic about his vork, being ready to put in long hours or extra time when
necessary. Call (1922) says there must be a love for the work so great, in those
engaged in research, that it will enable him to push forward in the face of obsta
cles which may seem insurmountable. (7) Courage: One should always have the cour
age of his .convictions. He should not be afraid to try something new.
( c ) Initiative in Experimental Projects
The project system has an enormous value in the coordination, continuity,
and conclusion of agricultural experimental work because it requires the submission
of an outline and its approval before any work is done. Success in experimental
projects depends upon the leader, his scientific attitude, depth of motive, concep
tion of the problem, and its requirements. Not all research is good. In fact,
there is a chance for much waste. . While partial failure is inevitable, it is possi .
ble for the investigator to gauge plausible success. Allen (1930)? advises research
workers to think scientifically, avoid adherence to routine, and keep abreast of the
times. The investigators should avoid the belief that his own compartment is water
tight and selfsufficient.
VT. Results of Agron o mic Research
Many contributions have been made in crops and soils by the experiment stations,
particularly during the past 25 years. Some of the more important advances in the
past quarter of a century in field crops are summarized by Warburton (1933) while
those in soils are given by Lipman (1933)*
(a) Field Crops
Among the contributions in corn have been the discovery that the show type
ear is unrelated to its performance in the field, that eartorow breeding may not
lead to improvement in corn yields, and that the combination of inbred lines in
hybrids has resulted in higher corn yields. In wheat, the discovery of rust resist
ance and of physiologic races has enabled investigators to breed for resistant
varieties. The seme is true for bunt. The introduction and use of sorghums, as well
as their improvement, has resulted in their production throughout the west. The : '
cause of flax "sickness" has been discovered as due to wilt with the result that re
sistant varieties have been bred. Sweet clover, once a weed, has been found to be a
valuable crop. Many improved varieties of crops have been developed for disease re
sistance, drouth resistance, high quality or high yield. Marquis wheat is one of the
most widely known improved varieties.
Tillage has been shown to be beneficial because of weed control rather than
moisture conservation from a dust mulch. Both Funchess (1929) and Richey (1937)
have given similar lists of advances made in field crop science.
(b) Soils
A quartercentury ago, physical chemical analyses of soil without other data
were frequently erroneous as a basis for the estimation of the agricultural value of
soils. In recent years some of the more valuable contributions have been as follows:
(1) Use of mineral fertilizers to Improve soil fertility; (2) ionic exchange in soil
colloids that led to an explanation of alkalisoil formation; (3) soil classification
and soil survey; (k) soil acidity in its relation to plant growth; (5) soil colloids
and their properties; (6) soil bacteria and other organisms end their influence on
soil fertility; and (7) soil erosion and its control. That soil productivity may be
maintained for a long period of time by the use of sound rotation and manurial prac
tices has been shown by the Morrow plots at Illinois. The results for 39 years have
been summarized by De Turk, et al . (1927).
VII . Value of Early Agronomic E xperiments '• ...
Some of the investigational work in agronomy "before 1910 was of little value due to
errors in the experiments , many of which were great enough to vitiate the conclusions.
Contradictions were common. As Piper and Stevenson (1910) point out, results were
sometimes suppressed "because they failed to coincide with current theory. "In short,
all scientific evils necessarily associated with experimental methods are too evident
in the field work in agronomy." The same type of criticism applies to other agri
cultural branches at that time. There were many reasons for this situation. The
"guess method" was widely used by the old school of experimenters for the accumula
tion of information. They usually lacked facte, lacked a broad outlook, were limited
in their experiences and, in many cases, had wide differences in viewpoints. Some of
the short coinings have been due to. pressure for information with the result that the
conclusions were often based on too few data. Other weaknesses were due to the view
point in some quarters that empirical facts were preferable to fundamental informa
tion from a practical standpoint.
While many of the early experiments would be inacceptable today in the light of
modern experimental standards, they nevertheless contributed to progress. Some of
them were as well conducted as those of today. Early agricultural practices were
determined quite as much by opinion as by experiment. It would have been a poor ex
periment indeed we're it to be less reliable than unsupported opinion. These early
experiments must be evaluated in relation to the knowledge of the time as well as
their effects on agricultural science and practice. For example., the early Sotham
sted investigations on the source of nitrogen in plants finally led to a solution of
the problem even tho the field experiments conducted in connection with them would
be considered today as inadequate.
Many of the weaknesses in early experiments have been met gradually through (l) wider
application of modern statistical methods, (2) replication of plots or treatments,
and (3) wider use of the inductive or scientific method in which general principles
are sought rather than empirical facts.
VIII . P resent Trends in Agronomic R esea rch
Some very definite trends are apparent in modern agronomic research, among them being
the emphasis on design of experiments, longtime projects, and regional coordination
of research.
(a) De sign of E xp eriment s
In recent years, a great deal of stress has been placed on the design of
experiments. The field layout and the method of analysis of the data are coordinat
ed so as to lead to more efficient experimental results. The emphasis on design has
been made by the Pothamsted workers. Design focuses attention on the objects of an
experiment that can be attained in no other way. This trend promises to reduce the
number of situations where an experiment is conducted and data collected before a
method of analysis is conceived.
(b) Longtime Pr ojects
Another definite trend is toward the longtime .project. According to Henry
Wallace (1936), "The solution of problems related to crop production is a matter of
years. The improvement of plants by breeding must extend through many generations.
Varieties must be compared in a number of different kinds of seasons for correct
evaluation. The same is true of tests of fertilizers, spraying practices, and cul
tural methods. To be productive, a program of plant research accordingly must be
stable, with a concentration of effort until a given problem is solved or its solu
tion found impractical for the time being."
(c) Regional Coordination of Res earch
The regional coordination of research work to reduce duplication of effort
is "being regarded more and more as essential. It has "been stressed "by Call ( 193*0 >
Jarvis (1931)* and others. Agronomic research "began as isolated hits of investiga
tion to solve local problems. Cooperation and coordination was developed later to
reduce wasteful duplication. It also makes possible a comprehensive attack on intri
cate problems, as well as the elimination of artificial boundaries. Such effort en
courages personal contacts and exchanges of ideas between different investigators.
Various bureaus of the U. S. Department of Agriculture took the leadership in region
al coordination. The most formal efforts on regional coordination are in the north
eastern states on pasture investigations and soil organic matter studies. The limi
tation of initiative and individuality of investigators lias sometimes been feared as
a result of regional coordination, but for the most part appears to be unfounded.
Referenc es
1. Allen, E. W. Initiating and Executing Agronomic Research. Jour. Am. Soc. Agron.,
22 : 3^1 . 1930.
2. Ball, C. R. Some Problems in Agronomy. Jo. Am. Soc. Agron., 3:3373^7. 1.916.
3. Call, L. E. Increasing the Efficiency of Agronomy. Jour. Am. Soc. Agron.,
Ik: 329339. 1922.
k. Regional Coordination of Agronomic Research from the Standpoint of
the Station Director. Jour. Amer. Soc. Agron., 26:8138. 193^.
5. Carleton, M. A. Development and Proper Status of Agronomy. Proc. Am. Soc.
Agron., 1:1721+ . 1907,
6. DeTurk.. E. E>, Bauer, F. C, and Smith, L. H. Lessons from the Morrow plots.
Ill.'Agr. Exp. Sba. Bui. 300. I927 .
7« Funchess, M. J. Some Outstanding Results of Agronomic Research. Jo. Am. Soc.
Agron., 21:1117. 1929.
o. Hall, A. D. An Account of the Rothamsted Experiments (Preface and Introduction)
1905.
9» Fertilizers and Manures, pp. I38. 1928.
10. Haskell, S. B. Agricultural Research in its Service to American Industry. Jo.
Am. Soc. Agron., l^tk^kQl. 1923.
11. Howard, A. Crop Production in India, pp. I85I95. 1924,
12. Jarvis, T. D. The Fundamentals of an Agricultural Research Program. Sci. Agr.,
26:8138. 193^.
13. Lipman, J. G. A Quarter Century of Progress in Soil Science. Jo. Am. Soc.
Agron., 25:925. 1933. '
Ik. Office of Experiment Stations. Legislation and Rulings Affecting Experiment
Stations, Miscel. Publ. 202. 193**.
15. Pearson, Karl. The Grammar of Science, P. 30. 1911.
16. Piper, C. V., and Stevenson, W. H. Standardization of Field Experimental Methods
in Agronomy. Proc. Am. Soc. Agron., 2:7076. 1910.
17. Richey, F. D. Why Plant Research. Jour. Am. Soc. Agron., 29:969977. 1937
18. Thorae, C. E. Essentials of Successful Field Experimentation. Ohio Agr. Exp.
Sta. Cir. 96. I909 .
19« True, A. C. A History of Agricultural Experimentation and Research in the United
States. Miscel. Publ. 251, 1937.
20. Warburton, C. W. A Quarter Century of Progress in the Development of Plant
Science. Jo. Am. Soc. Agron., 25:2536. 1933 .
21. Wheeler, H. J. The Status and Future of the American Agronomist. Proc. Am. Soc.
Agron., 3:3139. 1911.
b
Questions for Discussion
1. What conditions led to the subdivision of the agricultural field?
2. What are the functions of agronomy as a science? Why?
3 Who founded the first experiment station? What results were obtained?
k. When was the Rothamsted Experimental Station established? Where? By whom? Why?
5. What has Rothamsted contributed to early agricultural science?
6. Where was the first American experiment station established? When? Upon what
did It work?
7. Give some reasons to justify agricultural experimentation as a public duty.
3. Why Is agriculture in America a national concern second to none?
9. Give several reasons why a farmer is generally unable to do experimental work.
10. Name the acts of Congress that contributed to the agricultural experiment sta
tions, together with their dates of passage.
11. What special requirements is necessary for the expenditure of funds under the
Adams Act? Bankhead Jones Act? Hatch Act?
12. What kind of basic training Is necessary for agronomic research?
13. What would you consider as some of the most important attributes of a successful
investigator?
Ik, Discuss briefly the following characteristics in relation to research:
(l) imagination, (2) classification, {')) discrimination, {h) accuracy, and
( 5 ) t horoughne s s .
13 Why has the project system been useful in research?
lo. Name five contributions to crop knowledge made by experiment stations. Five con
tributions to soil science.
17. What are some reasons for the early contradictions in agronomic science?
18. What was the value of early agronomic experiments? What were some of their weak
nesses?
19 Name and discuss three trends in agronomic research at the present time.
CHAPTER II
HISTORY OF BASIC PLANT SCIENCES : . / . "
I. Early History of Basic Sc i ences
Agronomy as a science "began with the establishment of the first experiment station
by Jean Baptiste Boussingault in 183k, although many empirical facts were known be
fore that time. '.•
(a) Early Science
Science, in general, dates from Aristotle who was the founder of zoologj and
the forerunner of evolution. . A a one of the founders of the inductive method he
first conceived the idea of organized research. In fact, his principles might well
be observed at the present time. After Aristotle, little progress was made for 2,000
years. Among his theories was the one that the universe was composed of four ele
ments: air, earth, fire, and water. This was accepted for centuries because the
habit was to assume some man as an authority rather than to investigate. At the
beginning of the 17th century, Newton and Galilee began to base conclusions on facts.
Francis Bacon wrote books which emphasized that theories should be based on facts
rather than on authorities.
(b) Reasons for Slow Progr ess in Scie nce
Progress in agricultural science has had to wait on discoveries in the basic
sciences of physics and chemistry. There are many reasons for the slow development
of science in past ages, (l) Slavery was the general rule, with the result that
there was little stimulus to improve. (2) Experimenters lacked accurate instrument's
for measurement. (3) The mildness of the climate in the early civilized ceuntries
restricted industry, (k) Mathematical science was restricted. (5) The scientific
method developed by Aristotle was seldom used. Instead, it was the habit to assume
a general law. (6) Superstition and interference by the clergy discouraged experi
mentation.
II. Development of Agricult ural Science
There was little activity in the sciences related to agriculture before _l800. Funda
mental discoveries at the close of the l3th century, together with the appearance of
several treatises on agriculture, started .rap id development. 'Sir Humphrey Davy
(1813) published a book entitled "Essentials of Agricultural Chemistry" in which he
brought together many known facts. A vcn Thaer (l8l0) published a book on "Reasons
for Agriculture" in which he emphasized the value of humus in the soil, from which he
believed' plants gained their carbon. In 18U0, Justus von Liebig published hir: book or
organic chemistry in relation to agriculture, in which he advocated that the soil
need only be supplied with minerals.. This latter work struck the scientific world as
a thunderbolt. It has had a great deal of influence on modern agricultural research.
The establishment of the Rothamsted Experimental Station in I838 also ; reflected the
interest JLh agricultural science. The discoveries important to agriculture since
I85O have been: (1) The theory of evolution, (2) the discovery of anaerobic bacteria,
(3) the source of nitrogen in plants thru the aid of bacteria, (h) Mendel's laws of
heredity; (5) the chromosome theory of heredity as a physical basis for inheritance;
and (6) the discovery of vitamins. . .
•9
10
A — Plant Nutr ition
III. Early Plant Discoveries
Very little information was gathered on plant science from the time of the Greeks up
to the Renaissance, (l) Theophrastus : Published a "book on plants entitled "Enquiry
into Plants." He classified plants into herbs, shrubs, and trees. Theophrastus al
so distinguished bulbs, tubers, and rhizomes from true roots. Plant adaptation was
discussed. (2) Al Farbi: Discovered respiration in plants about 950 A.D. (3)
Johann van He lmont: This worker, who lived in the 17th centur^r, believed that water
was transformed into plant material . He placed 200 lbs . of soil in a receptacle and
grew a willow in it. Nothing was added but water. At the end of five years, he
found the willow weighed 169 lbs. and 3 oz., while the original soil lost only 2 02.
from its original weight. He concluded that the growth came from the water alone,
but failed to consider the air. (k) Jethro Tull : Believed that earth was the true
food of plants and that they absorbed soil particles. Therefore, he believed it
necessary to finely pulverize the soil through cultivation. Tull developed cultural
Implements and devised a system to plant crops in rows.
TV. Source of Nitrogen in Plants
The period from 18^0 to 1885 was taken up largely with the RothamstedLiebig contro
versy on the source of nitrogen in plants.
( a ) Earlier Work on Nitroge n
The element nitrogen was discovered in 1772. Joseph Priestly, followed by
Jans IngenHausz, settled the fundamental fact that green plants in sunlight decom
pose the carbon dioxide from the atmosphere, set oxygen free, and retain the carbon.
This source of carbon accounts for the bulk of dry matter in plants. From his work
in 1804, Theodore De Saussure concluded that plants were unable to assimilate free
atmospheric nitrogen, but obtained it from the .nitrogen compounds in the soil. The
pot experiments carried out by J. B. Bouesingault, who began his investigations in
l8o4, indicated that plants draw their nitrogen entirely from the soil or manure.
(b) LiebigRothamated Controv ersy
Justus von Liebig in lQkd maintained that green plants, by the aid of sun
light, derive their total substance from carbonic acid, water, and ammonia present in
the atmosphere, and from simple inorganic salts in the soil which are afterwards
found in the ash when the plant is burned. Liebig believed combined nitrogen in the
soil to be unnecessary in plant nutrition. This view was disputed by Lawes and Gil
bert who began elaborate experiments at Rothamsted in 1857. They grew plants under
glass shades, ammonia from the air being kept out. The earth, pots, manures, etc.,
employed in the experiment were burned to sterilize them. Carbon dioxide was intro
duced as required. Lawes and Gilbert made their trials both without manure and with
ammonium sulfate . Their work was done so carefully that the possibility of nitrogen
fixation by plants was excluded. While they concluded that plants require combined
nitrogen from the soil, they were unable to account for the gain in nitrogen in some
plants under field conditions. They. found actual gains in nitrogen when leguminous
plants were grown in the field, which was in agreement with the long experiences of
practical farmers.
(c) Final Experiments on Nitrogen Relations
The final experiment on nitrogen assimilation by plants was performed by
H. Helriegel and H. Wilfarth who found the symbiotic relationship between bacteria
and legumes. When he grew plants in sand, Helriegel (1.886) found that the Gramineae,
drucifereae, Chenopodiaceae, etc., grew almost proportionally to the combined nitro
gen supplied. When absent, nitrogen starvation took place as soon as the nitrogen
from the seed was exhausted. In legumes, he found that the plants were able to
11
recover and. begin luxurious growth. The. roots always had nodules on them in such
instances. However, legumes grown in sterile sand behaved the same as other plants,
but recovery could be brought about when a watery soil extract was added to them.
Renewed growth and assimilation of nitrogen was found to depend upon the production
of nodules on the roots. VFilfarth (1887) found bacteria in the nodules and settled
the point that bacteria are associated with nitrogen fixation. Later, these results
were confirmed at Rothamsted and final proof was obtained on the role of nitrogen in
plants. As Hall (3.905) recounts, the "very vigor" of the Rothamsted laboratory pre
vented fixation of nitrogen by the. exclusion of all possibility of inoculation. The
legumes as a class were found to be an exception to the contention that plants could
use only combined nitrogen from the soil. Both schools were partly right.
B  Evolution and Genetics
V. Early Work in Genetic s
Although many facts of inheritance were known previously, Genetics has been regarded
as a science only since 1900. At that time, the work of Gregor Mendel, originally
published in I865, was brought to light. Early work is reviewed by Roberts (1919;
1936), Zirkle (1932, 1935), and by Cook (1937).
( a) Sex in Plants
The Bisexual nature of the date palm was recognized by the early Babylonians
and Assyrians 5000 years ago. The ancients ascribed many monstrosities to hybridiza
tion. Many theories of heredity were in vogue, but no experimental data. Theophras
tus and Pliny discussed sex in plants, Primitive men made improvements in crops,
rice and maize being good examples.
About lbOO a new spirit of scientific skepticism began to be manifest. Many
of the cumulative absurdities and theories were being put to experiment. The in
creased interest in biology culminated in the publication of the famous letter by
Camerarius in 169^ on sex in plants. He gave convincing evidence that plants are
sexual organisms. Sex in plants was demonstrated by actual experiments with spinach,
hemp, and maize. ., .,;.,
(b) Hybridization of Plants
This work was followed by the production of the first artificial plant hybrid
by Thomas Fair child in England, a short time before 171?. In the next 50 years there
occurred a veritable wave of hybridizing. Crosses between more than a dozen genera
were made by several investigators. This period culminated in the publication of the
work of J. G. Koelreuter (I76I66) in which he reported the results of 136 experiment*
on artificial hybridization. In 1793* C. K. Sprengel observed cross pollination of
plants by insect 3. However, Zirkle (1932) calls attention to the fact that insect
pollination was observed by an American named Miller at a much earlier date. From
I760 to 1859 there followed many experiments on plant hybridization in attempts to
determine the nature cf inheritance. In 1822, John Go 3 s (England) reported but
failed to interpret dominance and. recessiveness, and segregation in peas. A. Sageret
(France) in 1826 classified contrasting characters in pairs, using muskme Ions and
cantaloupes. K. F. von Gaertner reported in 1835 on hybridizations made with 107
plant species. He noted plant vigor and the, uniformity of the first generation after
a cross. In 1863, C. V. Naudin (France) published a memoir on hybridization in which
he almost discovered the laws of inheritance.
VI . The Theory o f Evolution . . . ' \ : . '• . ■ ' • '••• " v .•'
The theory of organic evolution is one' of the most profound theories expostulated in
the past 300 or hOO years. It was brought to fruition in the publication of the.
"Origin of Species" \)y Charles Darwin in I859. Hybrids are discussed extensively,
12
but its contribution to genetics was mostly indirect. It marked the "beginning of
the modern experimental approach to biological problems.
(a) Evolution before Darwin
When Darwin published the "Origin of Species" spontaneous generation and
special creation were the current theories. A great majority of naturalists believed
that species were immutable productions specially created. Up to this time, empiri
cal rather than scientific improvement had been made in plants. Darwin did not
originate the evolution theory; he merely furnished evidence for its substantiation.
Aristotle had expressed the central idea of evolution. Modern philosophy from Fran
cis Bacon onward shows definiteness in its grasp and conception. Erasmus Darwin,
grandfather of Charles Darwin, had a theory similar to that propounded in the "Origin
of Species." J'.B.P. de Lamarck, in his "Philosophic Zoologique" published in 1809,
made the first attempt to produce a comprehensive theory of evolution. He added the
idea of "use and disuse." Lamarck believed in the inheritance of acquired characters
.and attributed some influences to direct physical factors. In other words, all the
principal factors of evolution had been worked out before the time of Darwin with
the possible exception of "survival of the fittest" which he obtained from a book by
Malthus on population. ...
(b) The Work of Cha r les Dar win
Darwin made an extended trip around the world in the Beagle, collecting
voluminous facts and making extensive observations , in support of his theory. He is
given credit for the evolution theory because he was the first to gather facts. He
attempted to show how and why new species arose. (1) Theor y of Natural S election:
Present organic forms are believed to have evolved from more simple forms in past
ages. The theory was founded on these facts: (a) Variations between individuals
are universally present; (b) a struggle for existence takes place between individuals;
(c) through natural selection these individuals with the most favorable variations
survive; and (d) heredity tends to perpetuate the favorable variations from natural
selection. (2) Reasons for Success : Darwin was successful because of his thorough
ness, accuracy, hard work, honesty, ability to see, and because he was a stickler for
details. He showed by example that disinterestedness, modesty, and absolute fair
ness are important attributes of character in intellectual work. Darwin (1859) him
self states that his success was due to a love of science, unbound patience for long
reflection on a subject, industry in the collection and observation of facts, as well
as a fair share of invention and common sense.
VII. The Cell in Relation to Inheritance
Independent progress was being made in other fields that were to have a profound in
fluence on genetics after 1900. A. von Leeuwenlioek (Holland) discovered the micro
scope and saw mammalian germ cells in 1677 The cell theory was propounded in
183839 by M. J. Schleiden and T. Schwann (Germany). This was the first generalized
statement that all organisms are made up of ceilsone of the greatest generaliza
tions of experimental biology. The union of sperm and egg cells, i.e., fertilization,
was first seen in seaweed by G. Thuret (France) in 1849 • A year later he showed that
the egg would not develop without fertilization. The chromosomes were described in
1875 by E. Strassburger (Germany). During the same year Oscar Hertwig (Germany)
proved that fertilization consists of the union of two parental nuclei contained in
the sperm and ovum. .W. F lemming (Germany) in 187982 describes the longitudinal
splitting of the chromosomes, and later observed (188485) that the halves of split
chromosomes went to opposite poles. Th. Boveri (Germany) in 188788 verified the
earlier prediction of A. Vfeismann that reduction in the ohrojir Eome stakes place. In
I898, S. G. Wavashin (Russia) discovered double fertilization in higher plants.
Thus, the physical mechanism of inheritance was pretty well worked out by the time
that the work of Mendel was discovered.
13
VIII. The Lavs of Inheritance . .
The turn of the century proved to he an epochal year in the experimental study of
heredity. The work of Gregor Mendel (Austria), an August inian monk, on Inheritance
in peas was rediscovered in 1900 "by Hugo De Vr5.es> C.F.J.E. Correns, and E. von
Techermak. The work had "been published originally in 1866.
(a) Discovery of Principles of Heredity
Mendel made crosses of peas and observed carefully the resemblances and
differences among different races. He began his work in 1857* The principles of
heredity which he put forth were as follows: (l) single heredity units, (2) allelo
morphism or contrasted pairs, (3) dominance and recessiveness, (h) segregation, and
(5) combination. The last two are generally recognized as the distinct contributions
of Mendel .
(b) Methods used by Mendel
There are several reasons for the success of Mendel . His work differed from
that of his predecessors in several respects. (1) He .made actual counts and kept
records of each generation. (2) One pair of factors was studied at a time. (3) His
material was carefully studied and selected. (K) He guarded against errors in acci
dental crosses. (5) He worked with large numbers. (6) The crosses were studied for
seven generations. Roberts (1929) comments as follows on the work of Mendel:
"Nothing in any wise approaching this masterpiece of investigation had ever appeared
in the field of hybridization. For farreaching end searching analysis, for clear
thinkingout of the fundamental principles involved, and for deliberate, painstaking,
and accurate followingup of elaborate details, no single piece of investigation in
their field before his time will at all compare with it, especially when we consider
the absolute absence of precedent and initiative for tho work.":
IX. Modern Developments in Genet ic3
The universality of Mendelian principles was verified' in plants, animals, and. man
within three years. In 1902, Hugo De Vries advanced the mutation theory to explain
sudden changes in plants that breed true, but which could not be accounted for by
Mendelian inheritance. He found sudden changes in the evening primrose to breed
true in certain cases. These mutations were believed to furnish the basis for evo
lution. This was soon followed by the pureline concept, i.e., variations in the
progeny of .a single plant of a. self fertilized species .arc not due to inheritance.
This was first put forth by W. L, Jchannsen (Denmark) in I905. H. TTilssonEhle
(Sweden) advanced the multiple fact or hypothesis in I908. The chromosome theory of
heredity was announced by T . H. Morgan in 1910. His gene theory included the prin
ciple of linkage of genes resident on the same chromosome. This brilliant hypothe
sis has been upheld in many experiments. Much recent work has been concerned with
polyploidy, the mechanism of crossingover, and sterility.
The principles of genetics have enabled plant breeders to make definite contributions
to improved varieties. Many new varieties are now grown on farms that have been
made possible through application of the laws of inheritance. Many varieties are
"made to order" to meet particular conditions.
— Other Basic Sc iences
X. Development of Bacteriology ..
Great advances were made in the field of bacteriology between 1.360 and i860, it be
ing definitely established that bacteria bring about putrefaction, decomposition,
and other changes. The work of Louis Pasteur dominated the field during this period,
Ik
(a) Pasteur and his Work
Pasteur discovered anaerobic "bacteria. Fermentation was commonly thought to
he the result of a chemical change, hut Pasteur proved it to he due to anaerobic
bacteria.. This wrecked the theory of spontaneous generation of life. Pasteur showed
that the presence of bacteria could always be traced to the entrance of germs from
the outside, or to growth already present. Other contributions of Pasteur included
the discovery of the causes of many bacterial diseases, and the development of
methods of immunization. Pasteur had several attributes that led to his success:
(1) He established truth by experiment; (2) He was discerning with regard to the
problem on which he worked; and (3) he worked on one problem at a time.
( b ) Other Di sco veries in Bacte riology
Many further developments in bacteriology depended upon the improvement of
the microscope and the perfection of various technics. The oil immersion lense was
developed about i860. The agar plate method for the study of growing colonies of
bacteria was introduced by Robert Koch in l88l. The transformation of ammonia to
nitrates was demonstrated by T. Schloesing and A,. Muntz in I877, but it remained for
S. Winogradsky to isolate the organisms concerned. That nodules are formed on
legumes as the result of inoculation with microorganisms was demonstrated by H.
Helriegel and H. Wilfarth in 1886. M. W. Beijernick isolated nonsymbiotic bacteria,
i.e., the Azotobacter, in 1901. Among other contributions of ■ bacteriology were
sterilization technics, the classification of bacteria on a physiological basis
(started by Ferdinand Cohn in I872) the study of diseases due to filterable viruses,
and studies in the nature of bacteriophagy .
XI. P lant Patholog y
Like all natural sciences, plant pathology had its start with the dawn of civiliza
tion. The Hebrews mentioned plant diseases in the Bible, but only gave descriptions
and mentioned damage. Little was known about plant diseases until the modern era
which began about I85O.
One of the greatest early workers was Anton de Bary (German) who proved the parasi
tism of Fungi in I853 . A little later (lQ6k) he proved heteroecism in rusts as
illustrated by the relation of the aecidium on the barberry to the red and black
rust stages on wheat . That bacteria may cause plant diseases was first proved by
Thomas Burrill in 187981 . He showed that a definite species, Bacillus amylovoris,
was the causal agent of fire blight. The use of Bordeaux mixture as a fungicide was
started in France in 1886. Since that time, many other fungicides have been used in
plant disease control, the latest being the organic mercury compounds.
Biologic strains in rusts were discovered in I89? 4  by J. Eriksson (Sweden), while
races within a variety of rust were demonstrated by E. C. Stakman and his coworkers
in I916. Another important discovery was made by J. H. . Craigie in 1927 when he dis
covered sexuality in the rusts.
A rapid increase in the knowledge of socalled virus diseases of plants has taken
place since the first proof of tobacco mosadc as an infectious disease in 1388. The
role of insects in the transmission of the virus or active principle was soon recog
nized. Recently, W. M. Stanley (1937) bas advanced strong evidence that the tobacco
mosaic virus is due to a high molecular weight crystalline protein.
A great deal of attention is now being given to the production of diseaseresistant
and immune varieties of crop plants through the application of genetic methods.
15
References
1. Cook,. B. A Chronology of Genetics. Yearbook of Agriculture, U.S.D.A.
pp. 14571477. 1937.
2. DampierWhetham, W.C.D. A History of Science, pp. 3340, l4ll47, IJQI69, and
285287. I93I. •■
3. Darwin, Charles. Origin of Species. 1859. •' /
4. Hall, A. D. The Book of the Bothamsted Experiments, preface and introduction,
pp. 114. I905.
5. Hall, A. D. Fertilizers and Manures, pp. I38. 1928.
6. Hayes, H. K., and Garber, B. J. Breeding Crop Plants, pp. ll4. 1927»
7. Heald, F. D. Manual of Plant Diseases, pp. 1720. 1933
8. Boberts, H. F. The Founders of the Art of Plant Breeding, Jour. Her., 10:99106,
147152, 229239, and 267275. 1919.
9. Boberts, H. F. Plant Hybridization Before Mendel . 1929.
10. Bus sell, E.J. Soil Conditions and Plant Growth, pp. 131. 1932.
11. Stanley, W. M. Crystalline Tobacco Mosaic Virus Protein. Am. Jour. Bot.,
24:5968. 1937.
12. Theophrastus . Enquiry into Plants, Vol. I and II (Translated by Arthur Hort) .
1926.
13. ValleryBadot, B. The Life of Pasteur. I928.
14. Weir, V. W. Soil Science, pp. 326. 1936.
15. Whetzel, H. H. An Outline of the History of Phytopathology.
16. Zirkle, C. Some Forgotten Records of Hybridization and Sex in Plants. Jour.
Her., 23:433448. 1932 .
17. Zirkle, C. The Beginnings of Plant Hybridization. 1935
Questions for Discussion
1. Who is considered the "Father of Science"? Why?
2. What were the contributions of Galileo, Bacon and Newton to science?
3 Why did science develop slowly previous to the l6th century?
4. Name several discoveries important to agriculture since 1850.
5. Who was Theophrastus, and what did he do?
6. What were the views of these men on plant nutrition: Von Helmont, Jethro Tull,
Thaer, Liebig?
7. What did deSaussure contribute to agricultural research? Sir Humphrey Davy?
Boussingault?
8. What facts made the source of nitrogen in plants so important a problem during
the 19th century?
9. Mention three theories that were proposed to account for the supposed extraction
of nitrogen by plants from the ail*.
10. Describe the experiments at Bothamsted conducted to determine whether or not
plants secure nitrogen from the air. What was the result of these experiments?
11. By whom, how, and when was the source of nitrogen of legumes discovered?
12. What important lessons are illustrated by the investigations relating to the
source of nitrogen in plants?
13 What did these men contribute to early plant science: Oamerarius, Kolreuter,
Sprengel, and Naudin?
14. Who originated the theory of evolution? Why was it not accepted at that time?
15. What was the status of plant and animal improvement at the time of publication
of the "Origin of Species"?
16. What did Darwin contribute to the theory of evolution? Why is he usually given
credit for it?
17. What is the theory of natural selection?
16
18. Describe the work of Mendel and tell why he was successful.
19. What is the mutation theory? Chromosome theory of heredity? Multiple factor
hypothesis?
20. What was the prevailing belief in spontaneous generation of life when Pasteur
began investigating the subject?
21. What are the principal contributions of Pasteur?
22. Name several advances made in bacteriology since the time of Pasteur?
23. Name 5 important discoveries in plant pathology.
l/
CHAPTER III
LOGIC HJ EXPERIMEOTATION
I. Scope of Science
Science is systematized knowledge. The function of science is the classification of
observations and the recognition of their sequence and relative significance. Its
scope is to ascertain truth in every "branch of knowledge., Sound logic is just as
fundamental to good science as accurate data. The thought process most important in
science is induction, i.e., reasoning from the particular to the general. General
izations may lead to laws and principles ahout natural phenomena.
II. Science among the Ancients
"Primitive peoples lived through thousands of years of myth and magic, while science
was rising out of slow and unconscious observations of natural events, "Weir (1936)
explains. Aristotle (38^322 B.C.), one of the first to stress science, taught that
it can he developed only through reason. He sot up a logical scheme, called the
syllogism, which severely limited deductions made from generalizations. Jevons
(I87O) describes the syllogism as follows: "In a syllogism we so unite in thought
two premises or propositions put forward, that we are enabled to draw from them or
infer, by means of the middle term they contain, a third proposition called the con
clusion." An example is as follows:
"All living plants absorb water; (major premise)
A tree is a living plant: (minor premise)
Therefore, a tree absorbs water." (conclusion)
The syllogism has been rejected for a long time because it lead3 to no new knowledge.
It involves a deductive process, the conclusions being only as accurate as the pre
mises upon which they are based. New generalizations can only be reached through
induction, a process which affords a means to attack the premises themselves.
A  Methods and Types of Research
III. Research
In the broad sense, the collection and analysis of data is research. However, there
are different degrees of research value. Black, et . al . (1923) state that the mere
accumulation of facts, computation of averages, or census taking is not research.
Fact gathering alone is a mechanical procedure unless tied up with analysis. More
over, projects designed to serve purely local or temporary' needs, without some con
tribution to fundamental principles, ordinarily can have but little scientific value,
General laws or principles are sought in research of the highest order. There are
two methods of research, the empirical and the inductive. Black, et . al . (1928)
state that "the essential difference between the two is that the first one accepts
superficial relationships without inquiry as to antecedents, whereas the second one
pursues antecedents a stage or two at least." An antecedent is a condition or cir
cumstance that exists before an event or phenomenon.
IV. Inductive or Scientific Method
The process of induction is of special importance in experimental science in which
general laws are established from particular phenomena. The inductive method is the
scientific spirit of the day.
17
13
(a) Explanation of Induction
In induction one proceeds from less general, or even from individual facts,
to more general propositions, truths, or laws of nature. In other vords, it is the
formulation of a principle from facts. Induction was the method of Francis Bacon,
•who held that general laws could he established with complete certainty by almost
mechanical processes. Bacon advised that one begin by collecting facts, classifying
them according to their agreement and difference. It is then possible to induce from
their differences and similarities the possible reasons for the relationships exhi
bited and, from them, arrive at laws of greater and greater generality. Thus, the
inductive method attempts to answer the question "why". A knowledge of causes
enables the scientist to forecast with greater and greater assurance because, when
he knows what is behind a set of relationships, he is in a much better position to
know whether or not they will occur again. On the other hand, deduction is the in
ference from the general to the particular, i.e., some truth may allow individual
facts to be subsummed under it. Induction and deduction are used together in ex
perimental work. For instance, a premature induction may be made to account for a
phenomenon. A hypothesis is set up that may or may not be faulty. Next, an experi
ment is designed, a purely deductive process, to test this hypothesis. The investi
gator determines the particular instances he may create and observe by experiment to
use as a basis of a re generalization to establish. the original hypothesis.
( b ) Observation
The first requisite of induction is experience to furnish the facts. Such
experience may be obtained by observation or experiment. Jevons (1870) makes this
statement; "To observe is merely to notice events and changes which are produced in
the ordinary course of nature, without being able, or at least attempting, to control
or vary these changes." The botanist usually employs mere observation when he ex
amines plants as they are met with in their natural condition. Progress of knowledge
by mere observation has been slow, uncertain, and irregular in comparison with that
attained in the controlled experiment. However, to observe well is an art that is
extremely advantageous in the pursuit of the natural sciences. One should make
accurate discrimination between what he really docs observe and what he infers from
the facts observed. The investigator should be ' : uninfluenced by any prejudice or
theory in correctly recording the facts observed and allowing to them their proper
weight", according to Jevons (1870).
( c ) E xperiment ation
In the experimental method in its pure form, a special hypothetical plan be
comes the basis of conclusions. The investigator varies at will the combinations of
things and circumstances, and then observes the result. Fisher . (193 7 ) describes
experimental observations as "only experience carefully planned, in advance, and de
signed to form a secure basis of new knowledge; that is,, they are systematically re
lated to the body of knowledge already acquired, and the results are deliberately
observed, and put on record accurately." In actual practice, the effect of differ
ent factors is determined by holding all conditions constant or uniform except the
one or ones whose "effects" are to be measured, a definite amount of change in this
condition being balanced against a definite amount of change in the result. Black,
et al, (l9?8) state that it is sometimes only the effect of the presence or absence
of a condition that, is noted. The method is qualitatively experimental instead of
quantitatively in such cases. In many cases, it is impossible to hold all conditions
but one constant or even uniform. So statistical analysis is combined with the ex
perimental design to measure variation where it cannot be controlled. This is the
practice in many agronomy experiments. For instance, when two or more wheat varie
ties are compared for yield, they are planted in the same field, at the same time,
and at the same rate. Moreover, they are harvested at the. same time, threshed by
the same machine, and the seed weighed on the same balance. The conditions arc thus
uniform for the varieties rather than constant. The importance of the experiment is
19
well summarized by Jevons: "It is obvious that experiment is the most potent and
direct mode of obtaining facts where it can be applied. We might have to wait years
or centuries to meet accidentally with facts which we can readily produce at any
moment in a laboratory . . . ."
(d) Essentials of Good Scientific Me thod
The essentials in sound experimental method may be briefly summarized as
follows :
1. The formulation of a trial hypothesis, v
2. A careful and logical analysis of the problem generated by the
hypothesis .
3. Use of the deductive method to design how to effect a solution
of the problem. This involves a detailed outline of the experi
ment with costs, equipment, methods, etc. The factors should be
expressed in quantitative terms when possible.
h. Control of the personal equation.
5. Rigorous and exact experimental' procedure with the collection of
data pertinent to the subject.
6. Sound and logical reasoning as to how the conclusions bear on the
trial hypothesis and in the formulation of generalizations. A
statement of the exact conclusions warranted from the cases exam
ined should be made in accurate terms.
T. A complete and careful report of data and methods of analysis so
that others can check them.
V. T he "Smpirical Method
"When a law of nature is ascertained purely by induction from certain observations or
experiments, and has no other guarantee for its truth) it is said to be an empirical
law," according to Jevons (I87O). Thus, knowledge is empirical when one merely knows
the nature of phenomena without being able to explain the facts. It only answers
the question "how". Formerly, the empirical method represented knowledge secured by
trial, but today it means the haphazard ''cut and try" method. A person who learns
certain facts through repeated observations may know no reason for their being true,
i.e., he cannot bring them into harmony with any other scientific facts. The method
is valuable in spite of the criticisms against it. Empirical methods are most likely
to be used when a science is new. Fact 3 must be gathered before a notion of reasons
can be formulated . The older crop rotation experiments were empirical. Recommenda
tions are based on the results, i.e., certain rotation systems result in higher crop
yields. Crop variety tests are generally empirical, since the chief concern is to
determine what variety yields the highest.* Fully onehalf the agronomic experiments
in this country are haphazard in the nature of 'their relationship to the body of
known knowledge in a given line. Too often they are not related to past experiments.
(See Allen, 1930) .
VI. General Types of Agr onomic Experiments
Agronomic experiments can be divided into field and laboratory or. greenhouse experi
ments. Questionaires and surveys are occasionally used to secure preliminary infor
mation. 
*Note: In recent years, variety tests may involve more than empiricism. Crosses are
often made to combine high yield with certain desirable quality factors or disease
resistance. The yield trial determines whether or not the result, has been accom
plished.
20
(a) The Field Exper iment
The field experiment involves the use of small plots, usually between l/lO
and l/lOOO  acre in size. The treatments are replicated, i.e., repeated on the
experimental area in tests designed to remove the error due to soil heterogeneity.
To make other conditions as uniform as possible, the varieties or treatments in the
experiment are treated as nearly the same as possible except for the factor or fac
tors under study. The field experiment has a wide application where yield is used
as a criterion to measure treatment effect. Field experiments may be classified as
follows: (1) Variety Test s: Such trials usually measure the yield of strains,
varieties, and species. Various combinations of forage crops for hay or pasture are
sometimes classified as variety tests. (2) Bate an d Date Tests : These experiments
are concerned with the yield response of a variety or crop when planted at different
rates or on different dates. (3) Crop d otation Tests : These trials include differ
ent series of rotations and crop sequences. jk) Cultural Studies : The time, manner,
and frequency of field operations are considered in such tests. (5) Fertilizer Ex
periments : These experiments usually include tests to determine the needs of nitro
gen, phosphorus, and potassium and their best combinations. Other considerations
are ways to supplement farm manures, value of cover crops and green manures, and the
amounts and methods of lime application. (6) Pas ture Experiments : Field experi
ments with pastures are generally used to study methods to seed and fertilize new
pastures, methods to renovate old pastures, and the influence of grazing on species
survival. (See Noll, 1928) .
(b) Laboratory and Greenhouse Experimen ts
Laboratory and greenhouse tests are often used to supplement field trials.
These tests often involve potometer and lysimeter studies as well as those based on
special techniques. Pot cultures are sometimes necessary for the study of the effect
of one factor by the exclusion of the others, or by their exaggeration. However, th^:
sole use of laboratory experiments may result in erroneous conclusions when applied
to field conditions. (S' ". Wheeler, 1907) • The use of laboratories and greenhouses
is on the increase because they have the advantage of controlled conditions. Some
agronomic problems adapted to such conditions are: (1) artificial rust epidemics,
(2) toxic effect of sorghums on crops that follow, (3) fertilizer cultures, (h) re
sistance of winter wheat to low temperatures, and (5) moisture, temperature, and
light relationship studies. Equipment for the study of hardiness in crop plants h&a
been described by Peltier (I93I).
Potometers are pots filled with soil in which plants are grown for experimen
tal purposes. To a greater' or less extent the earlier investigators assumed the
accuracy of such experiments when applied to field conditions. Lysimeters are modi
fied soil tanks used to measure the magnitude of nutrient losses from the soil by
leaching under various fertilizer and cropping conditions. Installation of lysi
meter equipment is expensive but permanent. The principal feature is the measure
ment of drainage water. A description of lysimeter equipment is given by Lyon and
Bizzell (1918) and by the American Society of Agronomy (1933)
( c ) Questionaires a n d Surveys
Very little use is made of either the questionaire or survey in agronomic
research. They are considered less desirable than the controlled experiment. The
questionaire consists of a set of questions to be answered without the aid of an in
vestigator (usually mailed) . It is impossible to secure accurate answers on ques
tions that are closely defined because the chances for misinterpretation are too
great. Survey data are collected with the personal aid of an enumera/tor or investi
gator. Spillman (1917) assumes that careful analyses of the methods of a large num
ber of farmers under essentially similar soil, climatic, and economic conditions,
may be made to reveal the success of one person and the failure of others. Ho found
that the discrepancy in the farmer's knowledge was small in large items, but increase"
21
as the importance to him decreased. Black, et al (1928) mentions some of the weak
nesses of the survey: (1) It does not furnish snough detail for some types of prob
lems; (2) It is not accurate enough for close analysis; and (5) It does not furnish
a large. enough sample for some purposes.
VII . Hypo these s, Theories, and Lavs
The difference between the hypothesis, the theory, and the law, is in the degree of
surety or the absolute.
a) Explanation of these Term s
When an idea is suggested by observed phenomena it is spoken of as a hypo
lesis. It represents a desire to explain the phenomena such as, for example, the
method by which plants take food from the soil. The hypothesis is important in the
deductive method in that, to best this preliminary induction, it is replaced more or
less completely by imagining the existence of agents which are thought adequate to
produce the known effects in question. Thus, Jevons (I87O) explains, the truth of
a hypothesis altogether depends upon subsequent verification. A theory is a limited
and inadequate verification of a hypothesis. Examples are the theory of the gene,
and the theory of evolution. A theory becomes a law when it is proved to be a fact
beyond a reasonable doubt. The Mendelian laws of heredity are good examples of laws,
(b) Formulation of a Hypot hesi s
There are certain advantages to the hypothesis: (1) It correlates facts;
(2) it forecasts other facts; and (5) it allows for discrimination between valuable
and useless information. Every experiment is the result of a tentative hypothesis
thought out in advance of the actual test. The hypothesis is based on the recogni
tion of coincident phenomena, or upon a familiarity with possible causes and effects.
Hibben (1908) states: "Hypothesis and experiment to Charles Darwin were like a two
edged sword which he employed with rare skill and effect." The hypothesis is the
precursor of the experiment which is merely an effort to solve the problem created
by the hypothesis.
( c ) / Qualities of a Good H ypothesis
There are several qualities that a good hypothesis should possess. These are
Allows: (1) It should be plausible. (2) It must be capable of proof, i.e., it
should provide a susceptible means to attack the problem created thereby (3) It
must be adequate to explain the phenomena to which it is applied, (h) It should in
volve no contradiction. (5) A simple hypothesis is preferable to a complex one.
There is little use to form a hypothesis on a complex basis unless it is possible to
collect the data by which it may be proved. A multiple hypothesis is made up of
several ideas. Occasionally it may be desirable to formulate several hypotheses.
Salmon (1928) advises an investigator to at least give consideration to all observ
able hypotheses. They are useful even though wrong because they eliminate that par
ticular idea from the problem. At any time, an investigator must be ready to aban
don a hypothesis or theory when further data prove the previous views untenable.
d) Null Hypothesi s
In all experimentation the null hypothesis is characteristic. The term has
been applied by E. A. Fisher (1937) in his "Design of Experiments ." The ba aid
a ssumption is th at no AAffnrnnr.R .CTist.fl hfitvsfin thfi t.T>ftn.t,rnftnt..q in the experiment,
i .e., they are samplfifl drown from the same general Eoj ailatiOH* Vnr instance, in a
variety test, the investigator makes the basic assumption that all varieties yield
alike. He can never prove this assumption but he may disprove it in the course of
experimentation. By the use of certain statistical arguments he may show a signifi
cant discrepancy from the hypothesis, i.e., the probability is that seme of the
varieties do differ in yield. Fisher (1937) states: "Every experiment may be said
22
to exist only in order to give the facta a chance of disproving the null hypothesis."
( e ) Crucial Tests
There may he two alternative conceptions or explanations which appear possi
ble. A crucial tost ( experiment urn crucis) is one by which two rival hypotheses can
be tested so that if one is proved, the other is immediately disproved. This is the
only means by which a hypothesis may be disproved. The first record of the applica
tion of the crucial test is attributed to Francis Bacon. A good example of a crucial
tost was the one applied by Richey and Sprague (1931) "to tost two theories for the
cause of hybrid vigor in corn which is expressed \tfien two inbred lines of reduced
vigor aru crossed to give the first generation hybrid. This additional vigor, Hichoy
(1927) explains, has been attributed to the physiologic stimulation hypothesis in
which heterogenous germplasm within the cells provides the stimulation. The other
hypothesis is that of dominant growth factors in which it is believed that the maxi
mum number of dominant growth factors are brought together in the first generation
hybrid, and that linkages of favorable dominant growth factors with other less de
sirable factors prevented the recovery of individuals as vigorous as the Fj_ in sub
sequent generations. Richey and Sprague (1933 ) applied a crucial test to the two
hypotheses by the collection of data on the principle of convergent improvement, i.e.
backcrossing the Fi hybrid to each of the two inbred lines that went into the hybrid.
It was hoped to transfer some of the favorable dominant growth factors from one of
the lines and intensify them in the other. Thus, the two convergently improved lines
would have less differences between them than was true of the original inbred lines.
Lowered yields of the cross of the convergently improved lines, as compared to the
cross of the original lines, would tend to support the physiological stimulation
hypothesis. The same or higher yields from the cross of the convergently improved
lines would lend support to the dominant growth factor hypothesis. The data collect
ed gave support to the latter.
B — Kinds of Evidence
VIII . I mpor tan ce of Evide nce
It is necessary to collect facts or data before generalizations can bo made. There
are different kinds of evidence, some kinds being more apt to lead to valid conclu
sions than others . However, plants are complex organic compounds with the result
that it is more difficult to determine the elements of cause end effect than is or
dinarily true in the more stable physical sciences. Environment has a tremendous
influence on the plant. The more that experiments or observations are repeated with
the same results, the more valid the evidence becomes in the minds of all normal
human beings. For example, a large number of experiments show that weed control is
the principal benefit derived from cultivation. The fact that a large number of
investigators have found this to be true under different conditions adds to the
assurance that the results are correct. Certain methods have been developed to deal
with the evidence obtained by observation or experiment which may serve as guides to
those in search of general laws of nature.
IX . C ause and Effect
Induction consists of inferring general conclusions from particular evidence. In
some cases, generalizations relate to cause and effect. An antecedent is a condition
which exists before the event or phenomenon, while a consequent follows after the
antecedents are put together. Jevons (I87O) makes this statement: "By the cause of
an event we moan the circumstances which must have preceded in order that the event
should happen. Nor is it generally possible to say that an event has one single
cause and no more. There are usually many different things, conditions or clrcum
2 5
stances necessary to the production of ah effect, and all of them must be considered
causes or necessary parts of the caused' It is certainly true that a multiplicity
of causes is often involved in experiments in field crops and soils.
X . Qualitative Evidenc e
Qualitative evidence is that which can "be measured only categorically. .For example,
seeds either germinate or failfar germinate. Classification "by color is a common
form of qualitative .data.
( a ) Method of Agreement
This method of induction is defined "by Jevons (I87O) as follows: "The sole
invariable antecedent of ' a .phenomenon is probably its cause." It is necessary to
collect as many instances as possible and compare together their antecedents. The
cne or more antecedents which are always present when the effect follows is consider
ed the cause. For example, when rust is present on wheat, low yields are obtained.
Therefore, rust causes low yiexds. This method has a serious difficulty in that the
same effect in different cases may be due to different causes.
(b) Method of Difference
In this method, the antecedent which Is always present when the phenomenon
follows, and absent when it is absent, is' the cause of the phenomenon when other
conditions are held constant . In ether words, when the circumstances are all in
common except one, i.e., the treatment, then the change that occurs is the effect of
the treatment. This is probably the most widely ured. method In experimentation. The
differences in crop yields under certain manurial treatments is an example of this
method .
(c) Jo int Metho d ■..■•■
In the words of Jevons (187O), the eioint method of. agreement and difference
"consists in a double application of the method of agreement,: .first to a number of
instances where an effect is produced, and'' secondly, to a number of quite different .
instances where the effect is not produced." For example, the experiments of Darwin
on cross and self fertilised plants may be cited, flo placed a net around 100 heads
to protect them from chance insect pollination. He als a placed. 100' : heads of tie
same variety where they were exposed, to bees. The protected flowers failed to yield
a single seed, while the unprotected 'ones produced 2';'20 seeds. Thus, cross fertili
zation by means of insect pollination was proved, to be a cause of seed set in this
case .
XI . Quantitative Evidence ■'•''• : .....
Every science, and "every question is first a matter of generalizations built upon
qualitative evidence. The effort to more firmly substantiate such generalizations
leads to the measuring of evidence quantitatively so that by. degrees, the evidence
becomes more and more precisely quantitative. .
(a) Method of Concomitant Variations : ' •' " v " ■..•" ...:> .
This method can be applied where the phenomena dan be measured. ■ Every degree
and quantity of the phenomenon adds new evidence in support of relationships that
exist between antecedents and consequents of the phenomenon. The method which em
ploys concomitant variations to determine the .degree of such relationship is called
correlation. For instance, an experiment with wheat results in a low yield under
conditions of heavy stem rust infestation, with variations to, the other extreme.
2)+
(t>) Method of Residues
There may be several causes, each of which produces part of an effect, and
where it may be desirable to know how much of the effect is due to each. This type
of evidence consists in the analysis of a given phenomenon to determine the residue.
For instance, manure contains something besides phosphorus, potash, and nitrogen as
shown by the residues. In plants it has been determined that other than the so
called 10 essential elements are used because analyses of the plant ash show others
to be present. The method of residues is constantly employed in chemical determina
tions .
XII. Relat ion to the Original Hypothesis
Some experiments fail in their objective in that there is insufficient evidence at
hand to permit the investigator to draw positive conclusions. However, this evi
dence is valuable. It has been called "negative evidence," but in reality there is
no such thing. Research would be much further along than it is today if all experi
ments had been reported in which the evidence was insufficient to prove the hypo
thesis that was originally set up by the investigator. Such evidence would have
saved other workers from a repetition of the work.
XIII. Us e of Analogy
Analogy is a form of inference in which it is reasoned that, if two (or more) things
agree with one another in one or more respects, they will probably agree in still
other respects. It is the simplest and most primitive form of evidence, its great
weakness being the fact that the cases compared may not be parallel. Analogy may be
tested by some inductive method. For example, the theory of evolution was suggested
to Darwin from the "Essay on Population" by Malthus . It suggested to him that the
struggle for existence is the inevitable result of the rapid increase in organic
beings. The idea necessitated natural selection or "survival of the fittest."
Another example might be cited in durum wheat. Durum wheat is adapted to Russia and
30 is Turkey wheat. Since Turkey wheat is adapted to the Great Plains in this coun
try, durum, wheat must be adapted to this region also. A common analogy made by
agriculturists is that crops can be improved by systematic selection because liv
stock breeders have succeeded in that way. Logic derived from analogy too often *
leads the inexperienced astray.
C  Methods of Discovery
XIV. Work of other Investigators
An investigator seldom takes up work today that is entirely new. He secures valuable
help from other research workers. The cooperative attitude among the workers on the
Purnell corn, projects is particularly commendable in this respect. They get together
occasionally to talk over their problems freely and to offer suggestions. They have
been unusually free with their preliminary data and unpublished results so far as
fellow workers are concerned. This attitude has done much to advance research in
corn improvement. The seed analysts have cooperated among themselves in a similar
manner. Scientific meetings result in a more or less free exchange of ideas to the
benefit of all. These gettogethers are a great aid and should be attended by re
search workers .
XV. Surprise s and Accidental Discoverie s
An important discovery is quite often made by accident. Several examples could be
Qited.
25
(a) Lemon Juice in Grasshopper Bai t
Some 25 years ago, two workers in the U. S. Department of Agriculture were
testing poison bran mash as a grasshopper "bait in Kansas. These men had oranges in
the lunch' they took to the field with them. While eating their oranges, some of the
Juice accidentally came in contact with the bran mash. The men noticed, that the
grasshoppers preferred, the mash that contained the orange juice. As a result of
this discovery, Kansas came out with the lemon juice formula in 1911.
( b ) Heterothalism in Stem liust ■
Prior to 1927 > it was believed that the pycnia on the upper surface of the
barberry leaf had no function. Craigie (1997) got the idea that the mycelium,
pycnia, and pycniospores of some of the pustules were plus sex strains and others
•ciinus sex strains. He happened upon the proof by chance. The first fly of the
season appeared in the greenhouse on May 17. He watched it idly as it sipped nectar
at one pustule and then at another. Professor Buller happened by and said at once:
: 'fhe solution of the problem is an entomological one. Copy the fly. Take the plus
pycniospores to the minus pycnia, and the minus pycniospores to the plus pycnia."
Craigie followed this advice by mixing nectar from different pustules. The pycnio
spores germinated' and brought on the development of aecia and aeciospor.es, the
diploid phase. He repeated his test many times and found it to be true. Craigie
proved his theory as follows: Flies were introduced, in some cages containing bar
berry plants with pustules on the leaves, while flies were excluded from other bar
berry plants. Aecia were formed in five days where the flies were present, but none
were formed where the flies were excluded.
XVI. Syst ematic Research
One of the principal methods of discovery is through systematic research where a
problem is attacked from all conceivable angles. An example is the contribution of
the Hawaiian Experiment Station on chlorosis. The pineapple industry was restricted
to a small area because of a discoloration of the foliage that showed it to lack
chlorophyll. The investigators on this problem first exhausted the possibilities of
disease, after which they analyzed the soil and. found it to contain considerable
manganese. Next, the workers used this highmanganere soil on soil that would grow
pineapples, and found that very little iron was taken into the plants. The "manganese
was thus found to inhibit iron absorption. The plants were then sprayed with iron
salts and the chlorophyll deficiency corrected. Pineapple trees are now sprayed at
the rate of 50 pounds of iron Baits per acre, the yield of fruit being doubled, as a
result .
XVII. Other Methods of Discovercy
Several other methods have resulted in significant discoveries. (1) Conflicting
Results : Disagreement between different research workers in their results often
leads to new discoveries. Pasteur became engaged, in a controversy with Leibig on
the spontaneous generation of life. As a result, Pasteur proved that all new life
arose from forms that had already existed'. Some of the most fertile fields for new
ideas are the first new hypotheses, theories, and ideas. (2) Acc ur ate Fork: Accur
ate work is necessary to secure dependable facts on which to base conclusions. More
information usually results from work done carefully than from that which has been
unplanned and carried out in a haphazard manner . In addition, the work of investi
gators must be accurate to withstand the close scrutiny of other workers and of
general opinion. Accurate work often has led to new discoveries. (3) Analogy: A
fruitful source of new ideas that sometimes leads to new discoveries is analogy. It
may suggest a hypothesis from the results' secured in other experiments. ' (k) Id eas
from Farmers : In agricultural research, the 'problems called to "the attention of ex
periment station workers by farmers i s an important source of 'discovery.
26
References
1. Allen, E. W. Initiating and Executing Agronomic Research. Jour. Am. Soc.
Agron., 22:3^1. 1930.
2. Black, John D., et al . Research Method and Procedure in Agricultural Economics
(mimeographed), pp. 120, 5890, 113126, and 298. 1928.
3. Craigie, J. H. Discovery of the Function of the Pycnia of the Rust Fungi.
Nature, 120:765767. 1927.
k. Fisher, R. A. The Design of Experiments. Oliver and Boyd. 2nd Ed. pp. 112,
and 1820. 1937.
5. Eibben, J. G. Logic: Deductive and Inductive, pp. lo9l82, 222277, and
291329. 1908
6. Jevons, W. Stanley. Elementary Lessons in Logic, pp. 9l6, 116117; 126135,
201210, and 2l827o. 1870 (reprinted in 1.928)..
7. Lyon, T. L., and Bizzell, J. A. Lysimeter Experiments. Cornell Memoir 12. 1918,
8. Noll, C. F. The Type of Problem Adapted to Field Plot Experimentation. Jour.
Am'. Soc. Agron., 20:^211^5. I.928.
9. Pearson, Karl. The Grammar of Science , pp. llpt. 1911*
10. Peltier, G. L. Control Equipment for the Study of Hardiness In Crop Plants.
Jour. Agr. Res., 1+3:177182. 1931 .
11. Richey, F. D. The Convergent Improvement of Seli'ed Lines of Corn. Am. Nat.,
61:1+301+1+9. 1927.
12. , and Sprague, G. F. Experiments on Hybrid Vigor and Convergent
Improvement in Corn. Tech.. Bui. 267. U.S.D.A. 1931.
13. Salmon, S. C. Some Limitations in the Application of Least Squares to Field
Experiments. Jour. Am. Soc. Agron., 15:225239« 1923
Ik. , Principles of Agronomic Experimentation. Kans. St. Agr. Col.
(Unpublished Lectures) . 1928.
15. Spillman, W. J. Validity of the Survey Method of Research. Dept . Bui. 529,
U.S.D.A. 1917.
16. Standards for the Conduct and Interpretation of Field and Lysimeter Experiments.
Jour. Am. Soc. Agron., 25:803828. I933.
17. Weir, W. W. Soil Science, pp. 1116. 1936.
18. Wheeler, H. J. Some Desirable Precautions in Plot Experimentation. Jour. Am.
Soc. Agron., 1:391+1+. I907.
Questions for Dis cus sion
1. What is science?
2. What is the syllogism? Give an example.
3^ Why has the syllogism been abandoned in experimental work?
k. What is research? Discuss different values of research.
5 How does the inductive method of science differ .from the empirical?
6. Why is it considered desirable to determine basic or fundamental lavs rather
than merely to determine what happens?
7. Distinguish between induction and deduction.
8. What part does observation play in research work? What precautions are necec
eary in its use?
9 What is an experiment? Discuss its use.
10. What are the principal steps in the inductive method of science? Which ones
are most often omitted?
11. Under what conditions is the empirical method justified?
12. Name some types of agronomic tests that are empirical in nature.
13. What serious limitation is true of the empirical method?
Ik. What are some reasons for criticism of the methods of research?
27
15. Classify field experiments and describe each class.
16. What place have laboratory and greenhouse tests in agronomic research?
17. How do questionaires and surveys differ?
18. Distinguish between potometer and lysimeter tests.
19. Distinguish between hypothesis, theory, and law.
20. Is it desirable to formulate hypotheses in experimental work? Why?
21. What qualities are necessary in a good hypothesis?
22. What is a working hypothesis?
23. What advantages are there, if any, in formulating multiple hypotheses?
2k. What is the null hypothesis?
25. What is a crucial test? Explain one.
26. Why is research often more difficult in plant sciences than that in the physical
sciences?
27. Distinguish between cause and effect.
28. Name, define, and illustrate five different kinds of evidence.
29. What is the most important inductive method in experimentation? Why?
30. What is analogy? Discuss its use and give an example.
31. What is the value of negative evidence?
32. Mention k ways in which discoveries are made.
33* How was the cause of chlorosis found in pineapples in Hawaii?
3^. Mention 3 discoveries and tell how they originated.
CHAPTER IV
ERRORS IN EXPERIMENTAL WORK
I . Types of Experiment al Error
Two kinds of error are common in experimental work, systematic errors and chance
errors. The investigator needs to be familiar with both kinds. Such errors should
be distinguished from mistakes and blunders. For example, a worker makes a mistake
when he puts down a weight of 10 lbs. when the scale actually showed the weight to
be 20 lbs.
(a) Systemat ic Errors
Systematic errors occur every time that an experiment is repeated, in the
same way. Most experimental plans involve some errors of this kind. For example,
suppose that a large number of winter wheat varieties are arranged systematically in
singlerow plots. Some of the varieties kill out because of lack of hardiness. The
varieties in the adjacent rows might yield abnormally high because of the additional
space from which they could draw moisture. Such an error would be repeated every
time the experiment is conducted in this manner. In this particular case, the com
petition effect could have been avoided by planting threerow plots for each variety
and only the center row harvested for yield.'
(b) Chance Errors
Errors which occur by pure chance with no definite assigned cause are known
as chance errors. They are generally small fluctuations due to minor causes. Chance
errors may accumulate to produce a sizeable deviation even though it be impossible
to foresee and analyze all causes that contribute to them. The principal reason for
statistical analysis in agronomic science is its very inexactness and the inability
to control chance errors. In case the present theory of plot technique is acceptable,
the variations in plot yields are due to chance errors and, in most cases, have been
found by experience to be normally distributed. This means that there are a large
number of small errors and a small number of large errors. Statistical methods are
employed in field, experiments to measure the effect of chance errors. In addition,
some systematic errors can be removed by these methods as will be shown later.
II. Sources of Error in Experimental Work
Evidence gained by experiment is disputed., according to Fisher (1937) either on the
grounds that the interpretation is faulty, or on the criticism that the experiment
itself is poorly designed. Errors are always possible and. seldom absent in experi
mentation.
( a ) Faulty Design and I nferior T echniqu e_
Experimental designs are inadequate or faulty when they do not afford a
proper opportunity for statistical analysis to analyze and measure experimental
errors, both chance and systematic. Fisher states: . "If the design of an experiment
is faulty, any method of interpretation which makes it out to be decisive must be
faulty too." The investigator may fail to take certain variable factors into account
Aside from these, various personal errors may have been introduced, such as careless
ness. Farrell (1913) lists a few sources of error in field experiments. Among the
controllable ones are. Incorrect weights of crop products, faulty determinations of
plot area, variations in quantities of products recovered and wasted, unobserved,
variations in field treatments, etc. Among the errors seldom controlled, he cites:
Plant variation, soil irregularities, uneven distribution of soil moisture, and tem
perature variations. Frequently, the total effect from all causes is great enough tc
influence the conclusions of the experiment. It might be added that some o^ these
errors can be measured and their influence on the conclusions removed.
28
2 9
(b) Improper Interpretation of Results
Two common types of misinterpretation of experimental results are drawing
conclusions from too few data, and carrying the Interpretation, "beyond the points
actually tested. (1) Conclusions drawn from too few data ; An experiment may be
inadequately replicated in time and space to Justify the conclusions drawn, Carleton
(1909) warns that some experiments are defective because they are run for an insuf
ficient length of time. Sometimes investigators are in too much of a hurry to ob
tain results. Another common mistake is to over emphasize small differences. Sta
tistical methods have done a great deal towards reducing invalid inferences due to
too few data. (2) Interpretation carried beyond points tested : Sometimes the in
terpretation of the results of an experiment is carried beyond the points actually
tested. Salmon (I923) believes that one of the chief sources of error in agronomic
literature is the tendency to generalize from experiments limited in their scope.
For instance, it should be quite obvious that laboratory tests may not always be
applied to field conditions. Such generalization must be justified by a similarity
of conditions. As an example, suppose phosphates were added to the soil in a fer
tilizer test in amounts of 100, ^00, and 600 pounds per acre. One would be unable
to draw conclusions on, , say 1000 pounds, because it is beyond the amount tested in
the experiment. It is obvious that a point may be reached where the addition may
have a depressive effect. Sievers (1925) points out that recommendations based on
variety tests conducted under different conditions as to soil, climate, and weather
than those under which the farmer operates are unsatisfactory.
III. The Personal Factor
Individuals differ greatly in the way they attack problems and carry out the various
details connected with them. For example, two men will seldom agree exactly when
they make measurements on the same thing because they do not "see exactly alike".
Such differences are apt to be more pronounced when personal judgment plays an im
portant role. The mixing, of materials illustrates a situation where individual work
ers may differ in the details of their procedures to an extent that the endproduct
is affected. Mechanical devices tend to do away with the personal factor.
When several individuals work on an experiment it is desirable for the same person to
complete an entire operation, or at least for all the treatments in a single repli
cate. For example, in a variety test the same person should plant the plots, harvest
them, and make the weights so far as possible. At least, the same crew should carry
out the details uniformly for all plots or treatments, preferably for the entire
test.
IV. Sources of Variation in Field Experiments
Certain limitations in plot work must he recognized. To quote Noll (1928): "The
most serious are that the experiments must be made under constantly changing condi
tions as to moisture and temperature, and that the average results for a given soil
in a given locality, no matter how carefully planned, are not necessarily applicable
elsewhere." The most common variations in field experiments are those. due to plants,
those due to differences in seasons, and those due. to the soil. The variations that
cannot be balanced out can be measured in a welldesigned experiment. Some of those
due to defined causes, such as. soil heterogeneity, can be removed or balanced in part
but not entirely. The variations that are due to unrecognized causes are measured
and assigned to experimental, error.
(a) Errors Related to the Plant ' \.
Variation may be introduced due to differences in acclimatization unless this
factor happens to be the one under study. Differences in stand may be a fruitful
source of variation, particularly in crops like corn, sorghums, etc., where plant
30
individuality is important. There are less corn plants on a unit area than wheat
plants. A further source of variation due to plants is the difference in moisture
content of the harvested crop. Correction to a uniform moisture basis is advocated
under such conditions. Plant competition may introduce still further error in plot
■results.
(t>) Variations in Seasons
Climate rather than soil may he the limiting factor in crop production.
Some varieties are known to withstand. extreme conditions like drouth or excessive
moisture "better than others. From uniformity trials with corn over' a 3year period,
Smith (1909) concluded that more variation in yield could he expected in seasons un
favorable for the crop. For that reason, tests conducted for only one or two years
may he very misleading. This situation may he remedied "by the extension of a variety
test over a number of seasons to determine the variety that thrives "best in an aver
age season. For a reliahle average of seasonal conditions, a variety test should he
conducted for at least three years and pref erahly more . Under dryland conditions,
it takes at least 10 years to secure a reliahle variety average. Variety comparisons
should he strictly comparable, i.e., compared only for the same years under test.
Usually this is accomplished hy expressing yields in percent of the standard or
check. Other factors that may cause the yields of varieties to vary from season to
season are: (1) The plots may he damaged by windstorms one year and not in another.
(2) Rodents may cause more damage in some years than in others. (3) Insects may he
troublesome in certain seasons, (h) Rust in small grains may reduce yields more in
some years than in others. (5) There may ho an inaccuracy in scale weights from one
season to another. (6) Carelessness in harvesting or threshing is another factor in
some seasons. (7) Sometimes the planter fails to drill out to the end of a plot with
a possible error in yield as a result. (8) Crooked rows may introduce errors in the
yields of row crops .
( c ) Errors due to Soil Var iation
It is impossible to secure a perfectly uniform soil for field experiments.
Differences in productive capacity commonly occur in different portions of the same
field. In fact, soils vary in composition arid productivity from foot to foot with
the result that it is impossible to say that any soil. Is uniform, even on small
areas. However, the investigator should secure as uniform a piece of ground as pos
sible. Sedentary soils are usually more uniform than drift soils, and level land
more likely to be uniformly productive than hilly land. Other factors that may in
troduce variation are: topography, under drainage, subsoil,; and previous soil
management practices.
V . Errors in Laboratory a nd Greenhouse Tests
There are many possibilities for error in tests of this kind. Probably the most
serious one is to draw conclusions from laboratory tests for field conditions with
out a field test. Laboratory bests should supplement, rather than replace the field
' experiment . «
( a ) Errors in Greenh ouse T e sts ' '
Some of the possibilities for error nay be listed as follows: (l) The number
of plants is small. Plant individuality assumes major importance, part icularly when
the investigator works with large plants. (2) There may be unequal distribution of
water. It is difficult to get a uniform distribution of water through a heavy soil.
(3) It is often important that the exact amount of water in a soil be known. This is
particularly true in pots for freezing tests. (U) There may be a lack of uniformity
in the soil itself. This may be alleviated by thoroughly mixing the soil in a
homogenous mass. The mixed soil should be packed uniformly in all pots. (f>.) Berne
insects may be restricted only to greenhouse conditions. As a result, the behavior
31
in the field may "be entirely different so far as insects are concerned. (6) There
may he a temperature or light differential under controlled conditions. A lack or
overbalance of either or both may introduce a systematic error in the experiment.
Le Clerg (1935), in a uniformity trial with ^00 small pots in a greenhouse experi
ment, found the per cent of dampingoff in sugar beets to be less in the border row
pots on a raised concrete bench than in those farther removed from the heat pipes.
The effect was almost absent in a bench provided with wall boards to deflect direct
heat. The unequal exposure to light or heat may be corrected in some instances by
rotation of the pot table periodically.
(b) Comparison of Potometer and Field Trials
Data from pot experiments and field trials were found by Coffey and Tuttle
(1915) to agree closely in fertilizer experiments. However,, many fertilizer analo
gies from pot tests have led to errors in interpretation. Kezer and Robertson (I927)
found no agreement between potometers and field plots in irrigation studies with
wheat. Potometers with late irrigation treatments became so dry that the soil pulled
away from the edge of the can. When water was added, most of it ran down the cracks
and out of reacn of the root systems of the stunted plants.
VI. Statistical Methods in Relation to Variation
The statistical method is the mathematical means to measure and describe variation
and to allocate its component parts to certain recognized sources. Variation can be
measured quantitatively thru the medium of an experimental design that takes into
account the recognizable sources of variation. The measurement of total variation
makes it possible to obtain a measure of that due to all uncontrolled sources. The
statistical method concludes its role when it gives the experimenter a means to com
pare the obtained quantitative measures of variation due to the recognized possible
causal factors with the variation classified as error and also with each other.
Thus, conclusions can be drawn in regard to the relative importance of the sources of
variation, :.:"..
VII . Classical Fallacies in Agronomy ■
A number of fallacies in agronomy have been listed by Salmon (1929). Many of these
ideas were accepted as facts until rather recently. An analysis of these fallacies
shows how each came to be accepted by. agriculturists. ; . ■
(a) Conservation of Moisture by the Dust Mulch
The effectiveness of the soil mulch in the conservation of soil moisture has
been under discussion for many years. The early work, on which the dust mulch theory
was based, was performed in the laboratory. Between I885 and 1900, King (1907)
shewed that the dust mulch was quite effective in the reduction of water evaporated
from the soil surface. In fact, the water loss was about onehalf that from a bare
soil. However, King worked in the laboratory with soil in tubes, the water table
being only 22 inches from the soil surface. On the basis of this and similar experi
ments has rested the conviction that the soil mulch would reduce evaporation losses
and materially aid in the conservation of moisture. This theory was believed and
practiced until tests by the Office of Dry Land Agriculture (USDA) proved that it was
without foundation. Call and Sewell (1917) showed that the soil mulch failed to in
crease the moisture in the soil. In fact, the mulched plots actually lost more water
than bare undisturbed soil. The limit of capillary rise from a free water surface is
only about 10 feet, according to the work of Shaw and Smith (1927) . However, they
found moisture losses to be quite rapid from unmulched soil where the water table was
h to 6 feet from the surface. Other experiments in Illinois, Missouri, and Nebraska
kave shown that corn yielded almost as much where the weeds were scraped with a hoe
as vVibtpo th/=^ plots were cultivated (mulched). Shaw (1929) reworked King's experiment
32
using soil tubes k feet high, and maintaining a constant water table at the bast of
each. The loss in the mulched tube was 38 per cent less than that from the tube in
which the soil was left bare. This test merely confirmed the fact that the results
from these soil tubes could not be applied to field conditions where the free water
surface is usually more than 10 feet from the soil surface. Under dryland conditions
where moisture conservation is ertrcmely important, the water table is very often
200 to J+00 feet from the surface.
(b) Deep Plowing for Mo isture Conservation
The theory that very deep plowing will save moisture by an increase in the
storage volume of the soil is an old one that dates back to about 1880. It was some
times advocated that the soil be stirred from Ik to 18 inches deep. Deep tillage
was widely advocated on the Great Plains along about 1910 by Hardy W. Campbell. Most
of the implements used were soon allowed to rust out in fence corners. Experimenta
tion very quickly showed that deep tillage (Ik to 18 inches deep) was impractical or
actually depressed the yields under dryland conditions. Brandon (1925) found. that
winter wheat grown on plots subsoiled every two years .actually yielded 1.3 bushels
per acre less as a 15year average than wheat on land plowed at ordinary depths.
Similar results were obtained in Wyoming by Nelson (1929).
( c ) C ontinuous Selec tion of Sma ll Gr a iris
It was believed at one time that continuous selection was a means to invaria
bly improve small grains. After 50 years of continuous selection, Vilmorin concluded
that no improvement had resulted in wheat, a self fertilized crop. The pure line
theory worked out by NillsonEhle and by. Johannsen showed that selection was effect
ive only in heterozygous material. This old idea on the value of selection was
probably due to a disregard of the difference between self and crossfertilized
plants.
( d ) Selection of Seed Corn by Sc ore  Card S t andar ds
Arbitrary score card standards were improvised in the early days as ideals
for seed selection in corn. These standards laid stress on such points as shape of
kernel, length of kernel, ears with wellfilled butts, and tips, percentage of grain
on the cob, weight of ear, etc. Uniformity of' jars was particularly stressed. The
height of the belief in the "pretty ear" was reach xi about 1910 when the most "per
fect" ear at the National Corn Show sold for several hundred dollars. When planted
in the field in comparison with ordinary ears, it failed to surpass them either in
yield or quality. This started a great amount of research on the relation of score
card points to yield. It was generally proved that such arbitrary standards are of
little value. In fact, close selection for type was generally shown to result in an
approach to homozygosity with a reduction in yield and vigor as a consequence. Some
of the investigators who aided in the. upset of this theory were: Cunningham (I9I6);
Love and Went z (I917); Olson, Bull and Hayos(l9l3) ; Kiesselbach (1922); and Richey
(1925)
( e ) C al c i umMagne s t urn Bat i o in Soils
A physiological balance seems to be necessary in nutrient solutions for a.
normal plant growth. In IS92, Loew proposed the calcium magnesium' ratio hypothesis.
He worked out the optimum ratio for a number of different plants in water cultures.
He concluded that either calcium or magnesium used alone was toxic, but that the
toxicity disappeared when these elements fell within certain limits. The ratios
which Loew used varied from 1 CaO : 1 MgO to "( GaO : 1 MgO . A large amount of inves
tigation has been conducted on this ratio in which it has been shown that a rather
definite ratio of CaO to MgO Is required in nutrient solutions for optimum plant
growth. The same applies to other nutrient elements" as well. However, there appears
to be little evidence to support the necessity for a definite ratio of CaO to MgO in
soils. Recently, Moser (1935) reported that the ratio itself showed no relation to
33
crop yields. The "beneficial effect of lime added to the soil was attributed to the
increase in replaceable calcium rather than to an alteration of the calciummagnesium
ratio. It is sufficient to state that Loew conducted his experiments with water cul
tures which probably react differently from soils.
. (f ) Addition of Burnt Limestone to the Soil
It is still believed by some farmers that the addition of burnt limestone to
the soil results in a destruction of organic matter and an increase in the soil acid
ity. That burnt limestone increased the acidity was reported by the Pennsylvania
Experiment Station. The theory, as taught, was based on small analytical differences
in soil analyses.
(g) Acid Phosphate and Soil Acidity
The use of green manure and acid phosphate was at one time said to increase
soil acidity. Grass and green material were known to decay and give an acid under
laboratory conditions. Careful work under field conditions has shown that bacteria
use up the organic acid formed. Acid phosphate was thought to increase soil acidity
because of the name. It has been changed to superphosphate recently for psychologi
cal reasons.
References
1 Brandon, J. F. Crop Rotation and Cultural Methods at the Akron Field Station.
Dept. Bui. 130U, USDA. 1925 .
2. Call, L. E., and Sewell, M. C. The Soil Mulch. Jour. Amer. Soc. Agron. 9:^96l.
1917.
3. Carleton, M. A. Limitations in Field Experiments. Proc. Soc. for Agri . Sci.,
pp. 5561. 1909.
k. Coffey, G. N., and Tuttle, H. F. Pot Tests with Fertilizers Compared with Field
Trials. Jour. Am. Soc. Agron., 7:128135* 1915.
5. Cunningham, C. C. The Relation of Ear Characters of Corn to Yield. Jour. Amer.
Soc. Agron., 8:188196. 1916.
6. Farrell, F. D. Interpreting the Variation of Plot Yields. CIr. 109, BPI, USDA,
pp. 2732. 1913.
7. Fisher, R. A. Design of Experiments, pp. 112. 1937.
8. Kezer, A., and Robertson, D. W. The Critical Period of Applying Irrigation Water
to Wheat. Jour. Am. Soc. Agron., Vol. 19, No. 2. I927.
9. Kiesselbach, T. A. Corn Investigations. Nebraska Agr. Exp. Sta. Res. Bui. 20.
1922.
10. King, F. H. Physics of Agriculture. 1907 .
11. Le Clerg, E. L. Factors Affecting Experimental Error in Greenhouse Pot Tests
with Sugar Beets. Phytopath., 11:10191025. 1935
12. Lipman, Chas . B. A Critique of the Hypothesis of the LimeMagnesia Ratio.
Plant World, 19:83105, and 119135.' 1916.
13. Love, H. H. and Wentz, J. B. Correlations Between Ear Characters and Yield in
Corn, Jour. Amer. Soc. Agron., 0:315322. 1917 .
Ik. Moser? F. The Calcium Magnesium Ratio in Soils and It 3 Relation to Crop Growth.
Jour. Amer. Soc. Agron., 25:265377. 1933.
15. Nelson, A. L. Methods of Winter Wheat Tillage. Wyo. Agr. Extd . Sta. Bui. lol,
1929.
16. Noll, C. F. The Type of Problem Adapted to Field Experimentation. Jour. Am. Soc,
Agron., 20:^211+25. 1928.
17. Olmstead, L. B. Some Applications of the Method of Least Squares to Agricultural
Experiments. Jour. Amer. Soc. Agron., 6: 190204, 191U.
3h
18. Olson, P. J., Bull, C. P., and Hayes, E. K. Ear Type Selection and Yield in
Corn. Minn. Agr. Exp. St a. Bui. l*jk. 1918.
19. Pichey, F. P. Corn Judging and the Productiveness of Corn. Jour. Amor. Soc .
Agron., Vol. 17, No. 6, 1925 .
20. Salmon, S. C. Principles of Agronomic Experimentation (Unpublished lectures)
Kansas State College. 1929.
21. Some Limitations in the Application of the Method of Least Squares
to Field Experiments. Jour. Amor. Soc. Agron. 15:225239. 1923*
22. Shaw, C. F. When the Soil Mulch Conserves Moisture. Jour. Amor. Soc. Agron.,
21:11651171. I929.
23, , and Smith, A. Maximum Height of Capillary Pvise Starting with Soil at
Capillary Saturation. Hilgar&ia, 2:599409. 1927
2k. Si overs, F. J. Outstanding Weaknesses in Investigational Work in Agronomy.
• Jour. Am. Soc. Agron., 17:8869. 1925 .
25, Smith, L. H. Plot Arrangement for variety Experiments with Corn. Proc. Am. Soc.
Agron., 1:8439. 1909.
Que sticn s_ for 1 )is ci'.asiori
1. Distinguish between chance and systematic errors.
2. What errors in field experiments can "be controlled?
3. What kinds of errors in field, experiments are not controlled? How are they mini
mized?
h. What errors can be made in the interpretation of experimental results?
5. How may the personal factor influence experimental results?
6. What are the general sources of variation encountered in field experiments?
7. What factors cause plot yields to differ from season to season?
8. What errors may occur in greenhouse tests?
9. How did the soil mulch theory originate and, in the light of present knowledge,
how might the error have been prevented?
10. Is there any experimental or scientific basis for the belief that very deep plow
ing (10 inches or more) is profitable? Explain how this idea originated,
11. How did the belief that good seed corn is characterized by deep, rough kernels,
and cylindrical ears originate?
12. What was the basis for the belief that a certain calciummagnesium ratio was
necessary for plant growth?
13. Explain the origin of the idea that burned lime decreases organic matter in the
soil .
lU . Is there any reason to believe that acid phosphate or green manure increases
soil acidity? Why was it thought they did?
15. Make a general statement which will explain the sources of error that have
o c cur r e d in agr onomi c s c i e nc e .
FIELD PLOT TECHNIQUE
Part II
Statistical Analysis of Data
CHAPTER V '"■'•
, . FREQUENCY DISTRIBUTIONS AND THEIR APPLICATION
I Measurements and Collection of Data
Quantitative data, collected as a result of measurements, are widely used in research
■work. To measure a quantity is to determine "by any means, direct or indirect, its
ratio to the unit employed in expressing the value of that quantity. (Weld, 1916) .
Every measure has some sort of linear scale, either straight or curved, on which the
magnitudes are read. This is because the human eye can measure length far more ac
curately than it can most other magnitudes. However, the investigator should realize
that there is no such thing as an exact measurement. Seldom will a reweight or re
measurement give exactD.y the same quantity because of inaccuracies that arise from
imperfect apparatus and judgment in estimation. An observer may tend to overesti
mate, or his measurements may be prejudiced, or his judgment may fluctuate. Because
it is next to impossible to arrive at a true value, measurements should be made as
carefully as possible in order to obtain the closest approximation. The units of
measure will depend upon the degree of precision required in the worlc. One should
distinguish between errors and inaccuracies due to carelessness. These are more
properly called mistakes. They consist of blunders like reading the wrong number on
the scale, recording a figure in a notebook wrong, forgetting to deduct tare, etc.
It is much easier to check the accuracy of weights when they are made more than once.
Sternal vigilance and care are necessary to reduce mistakes to the minimum. The in
vestigator should realize that it is impossible to evolve sound results from unsound
or carelessly collected data merely thru the application of a formula.
II. Statistics in Experimental Work
After data are collected, it becomes desirable to describe them, interpret them, and
induce from them. This is the realm of statistics. = , .■'■•■■■.".
(a) Statistics Defin ed
Statistics may be regarded as the mathematical analysis based on the theory
of probability applied to observational data in an attempt to summarize and describe
them so that conclusions can be drawn concominr the phenomena that supply the data.
Fisher (193^) states that the original meaning of statistics suggests it was a study
of populations of human beings living in political union. The methods developed,
however, have little to dc with political unity. In fact, they are applied to popu
lations, animate or inanimate.
(b) Use of Statistics .:.• ■ ■;
Statistics are used in astronomy, biology, genetics, education, psychology,
and many other fields. They aro particularly applicable to data concerned with life
or the products of life. Probably 75 to 80 per cent of the agronomic workers in
agricultural experiment stations use statistical methods, although only about one
half of these apply statistics to other than yield data. However, statistical methods
are being used more extensively as time goes on. ■:.'.•
III. Some Typical Statistical Term s
The effort to characterize and describe the data mathematically leads to the calcula
tion of various statistics ..The simplest of these is the average or mean. It is
natural for the first step to be an attempt to find a single measure which will best
describe the sum total of the information expressed In a mass of data. The best
single measure is the mean. However, it fails to tell the entire story. Among the
other statistics are the median, mode, average deviation, standard deviation, coeffi
37
38
cient, and correlation ratio. Among the derived values from these stat:l"tic, t : are
their standard errors and probable errors employed in the important problem of esti
mation and prediction.
(1) By a variable is meant any organ or character which is capable of variation
or difference in size or kind. This difference may be measurable as in height, tem
perature, weight, etc., or indirectly as in the case of color, occupation, etc.
Variation may be continuous or discrete*. For example, a temperature change from
60 to 6l degrees must pass continuously through every intermediate state between
60 and 6l degrees. On the other hand, variation may take place by integral steps
without intermediate values, as in population which can never go up or down by less
than one. (2) A variate (x) is an individual value of a variable, e.g., 3 feet,
200 grams, 15 pounds, etc. (3) The frequency (f) is the number of times a particular
variate (x) occurs between two limiting values of a variable, i.e., the number of
variates in any one class, (k) A population is the totality of individuals which are
to be studied with regard to a character and may be finite or infinite. (5) A sample
may be all or a part of a population. A random sample is a sample taken in such a
way that all individuals which make up a population have an equal chance of being
included in the sample .
IV. Rules_ for Computat Vn
It is desirable to be consistent in the number of decimal places used in computations,
and in the manner of dropping decimals. Suppose it is desired to retain two decimal
places. For a number like 82.575; the value can be made 82.58 by raising the odd
number to an even number. However, when the digit in the third decimal place is
greater than 5> the number is added, but dropped when it is less than 5 For the
square root of a quotient to be accurate to two decimal places, it is recommended
that the quotient be carried to four decimal places. This is especially important
where the square root is to be used in multiplications for other computations.
V. Arithmetic Average or Mean
Masses of unorganized data explain little or nothing. Individual measures are less
significant than a typical value which stands for a number of measurements. An
average or mean is such a value. It is the single constant most commonly employed to
describe the sample.
(a) Simple Arithmetic Mean
The mean may be considered the center of gravity of a sample. It is equal to
the sum of the individual measurements divided by their number.
x = x + X2 4 X3 + x n or x = Sx (l)
N N
where Sx = the sum of all the variates, H = the total number of variates, x = the
arithmetic mean, and xi, ^ . . . .x n . the individual variates.
For example, the yields of Golden Glow corn on J plots were 8V. 8, 86.9, and 89.9
bushels per acre. The arithmetic mean would be:
x = 8^.8 + 86.9 + 899 = 87.2
5
*Note: This usage is somewhat different than that in Genetics where a discontinuous
variation refers to a germinal change that breeds true, while a continuous variation
applies to variations due to environment and nonheritable.
Hi
39
(b) Mean of Beplicated Variates 1
It must be remembered' that the weight of each variate must be equal in the
sample. When certain variates .are repeated, the computation may be shortened by
merely considering each distinct variate multiplied by the number of times it appears.
Suppose 7 corn plants of variety "A" were measured for height in the first replica
tion and were found to average 59 inches. In the second replication, 3 plants were
measured, and averaged 67 inches. A total of 20 plants were measured for height in
a third replication and found to average 54 inches. Suppose one desired to know the
average height for the variety. A simple arithmetic mean of 59> 67, and 54, (i« e >
60 inches) would be incorrect because a different number of plants made up the origi
nal means in the different replications. The mean must be calculated so as to give
due weight to each variate for the number of times that it occurs. For instance, the
mean may be calculated as follows:
* = (59 x 7) + J2k * 20) x ( 6 7 x 3) = 169^ = 56.47 in.
30 . "30
The same result may be obtained by the addition of the original yd variates and
dividing by 30.
VI . The Frequency Distribution .
The mean for replicated variates may be calculated from a frequency table which is a
simple device by which a considerable quantity, of data may be organized in condensed
and classified form. Some data presented by Goulden (3937) on the yields in grams of
400 barley plots will be used to illustrate the frequency table. The yields which .
follow represent an aggregate of data in which there are 400 variates. Each measure
ment is a variate, i.e., a particular measured value of the variable (x) yield.
Yields i n Grams of 400 Squ are. _Yar d_ Plots o f Barley^
135 162 I36 157 l4l 130 129 176 171 190 157 1^7 176 126 175 13^ I69 I89 180 128
169 205 129 117 144 125 165 170 153 186 164 123 165 203 156 182 164 176 176 150
216 154 184 203 166 155 215 190 164 204 194 148 162 146 174 185 171 181 158 147
165 157 180 165 127 186 133 170 134 177 109 169 128 152 165 139 146 144 178 188
133 128 161 160 167 156 125 162 128 103 116 87 123 143 130 119 141 174 157 168.
195 180 158 139 139 168 145 166 118 171 143 132 126 171 176 115 165 147 186 157
187 174 172 191 155 169 139 144 130 146 159 164 160 122 175 156 119 135 116 134 ■•',
157 182 209 136 153 160 142 179 125 149 171 186 196 175 189 214 169 166 164 195
189 108 118 149 178 171 151 192 127 143 158 174 191 134 188 248 164 206 135. 192
147 178 189 141 173 187 167 128 139 152 167 131 203 231 214 177 161 194 141 161
124 130 112 122 192 155 196 179 166 156 13'L 179 201 122 207 189 1.64 131 211 172
170 140 156 199 181 181 150 184 154 200 I87 169 155 107 143 145 190 176 162 123
189 194 146 2£ 160 107 70, "34 112 162 124 136 138 101 138 l4l l43 135163 1S3
99 118 150 151 33 136 171 191 155 164 98 136 115 168 130 111 136 129 122 120
179 172 192 171 151 142 193 174 146 180 140 137 138 194 109 120 124 126 126 147
115 148 195 154 149 139 163 118 126 127 139 174 167 175 179 172 174 3.67 142 169
122 163 144 147 123 160 137 161 122 101 158 103 119 3.64 112 57 ' > (§3) 106 132 122
164 142 155 147 115 143 68. 184 183 167 160 138 191 153 loO 156 122 111 153  ] 43
103 131 180 142 191 175 146 101 ••111 ■• 110 154 176 168 175 175 146 148 167 106 123
121154 148 91 93 74 113 79 131U9 96 86 97 98 106 107 69 86 94 129
/r " • ■ :\ ':.,' Mi
**
lThis has been sometimes called a "weighted" mean. ;■'■'.•;
2 Data from Methods of Statistical Analysis by C. H. Goulden, p. 7, 1937.
ko :
(a) Grouping of Data into Classes
The above data are unwieldy in their present form, even though quite. simple
in nature. They may he condensed by grouping. First, find the highest and lowest
values of the variates (barley yields in grams). The interval thus defined by these
extreme values is known as the range. In this case it is 22 to 2l+8. The next step
in the formulation of a frequency.'  table or distribution is to separate the range into
classes. Although unnecessary, it is usually convenient for the classes to have
equal range (interval) within themselves. The number of classes to be formed is the
next question. Experience has shown that, somewhere between 7 and 20 classes is a
desirable number with which to work. The smaller the number of classes the greater
is the error due to grouping. The approximate number of classes can be determined
from a formula given by Yule (1929) :
Number of Classes = 2,p yNumber in Sample = 2.5 hJkOO = 11.18
Suppose 12 classes are decided upon. The quotient of the range divided by the number
of classes is the approximate class interval, viz,,, 226/12 = 18. ' However, a class
interval with an odd number is more convenient because the midpoint of the range does
not require an additional decimal. Suppose 19 is selected as the class interval.
The value of a class is taken at its mid value . The barley data may be tabulated for
a class interval of 19 as follows:
Class
Range
(gm)
Class
Value (x).
(gm)
. I ( & 
M£
2J
Tabulation
Frequency
(f)
(No.)
221+0
kl 59
6078
7997
98116
117155
13615,1+
155173
17I+192
193211
212230
23121+9
31
50
6q
83
107
126
li+5
161+
183
202
221
2 1+0
1
1
1111
itia mi 11
IHl 1111 IHI 1111 Kbil mi 1
(This tabulation can be
continued in Like maimer
for the other variates.)
1
1
1+
12
31
69
80
97
78
21
k
. N = 1+00
(b) Frequency Table
After the data are tabulated they are next arranged 'in a frequency table,
i.e., the frequencies are entered to correspond to their class values.
x
IX
31
1
31
50
1
50
69
k
276
38
12
1,056
107
31
3,317
126
69
8,691+
ll+5
80
11,600
161+
97
15,908
I83
73
Ik, 27k
202
21
k,2k2
221
1+
S3h,
21+0
2
1+80
H = 1+00 S(fx) = 60,812
The mean (x) for this sample can be con
veniently calculated from the frequency
table. Each class value is multiplied by
its frequency (f) to five fx. These values
are summed to give S (fx) and divided by
the total number in the sample. For the
barley yi
H o"!
G.lCLS ■
X
60 ,812 ==
"1+00
.Vii
■'O
S(fx) ; =
N
It should be evident that the classification
of the data into a frequency distribution ha?
distorted them from their original form.
kl
■ (c) Graphical Representation of Frequency Table .
A visible representation of a large number of measurements is afforded "by
either a histogram or a frequency polygon.
The histogram is most commonly used. The character to be measured is repre
sented along the horizontal axis (abscissa), while the frequencies are represented
vertically (Ordinate) to correspond to each class. For example, the barley yield
data may be plotted as follows:
■
f
100
80
60
ho
20
Frequency
(ordinate)
— i* —
. .
L^h— r~
T7l~i
31 50 69 88 107 126 1U5 16 J + 183 202 221 2^0
Class Values in Grams (abscissa)
x
The frequency polygon is constructed by joining in sequence the midpoints of the
tops of the bars of the histogram. Its shape tends towards the smooth curve of the
population from which the sample was drawn. The frequency polygon for the barley
yield data is as follows:
100
80
60
Frequency
(ordinate)
ho
20
X
31 50 69 88 10? 126 1U5 l6k 183 202
Class Values in Grams (abscissa)
221 240
VII. Measures of Central Tendency
There are three measures of central tendency that must he defined at this point.
(1) The arithmetic mean, already discussed, is the center of gravity of the popula
tion. (2) The median is the measure of the middle variate in an ordered arrangement
of the variates according to magnitude. (3) The mode is the measure of the class of
greatest frequency, or the point at which the most variates occur. In other words,
it is the xvalue at which the frequency polygon has the highest ordinate.
VIII . Types of Frequency Distr ibutio ns
Before one goes further with the analysis to describe the nature of the aggregate of
the data, it is necessary to roughly determine the type of frequency distribution.
Some mathematical expression is essentia.! corresponding to those types most often en
countered in actual practice. (1) A great many frequency distributions found in
practice are unimodal, i.e., have one peak. (2.) There is a general tendency for them
to be hell shaped when the frequency polygon or diagram is smooth. It was early
noticed that the curve derived from tin., theoretical distribution of the expansion of
a "binomial, (a < h) n , possessed many of the same characteristics of frequency distri
butions met with in actual practice. However, the "binomial distribution fails to
represent continuous variation. An effort to find a mathematical equation for a
curve which would well fit the points of a binomial d:i stribiat ion led to the discovery
of what is known as the normal probability curve and its equation. Types of distri
butions most commonly approached in thu graphical representation of data are the
normal, binomial, and the Poisson distributions.
(a) NormaJ Distributi on
The normal curve is a bellshaped, symmetrical curve. It is characterized
by the symmetrical arrangement of the items around the central value. The arithmetic
mean, median, and mode coincide in the norma,! curve. As in the case of mam'  frequency
distributions, the small deviations from the central value (mean) occur more frequent
ly and the larger deviations less frequently. Fisher (193*0 gives the statures of
3 375 women in a. curve that closely approaches, a normal curve.
Number in
each group
55
200
/T\
150
/ ■
100
J: \
\
50
/ I
^y \
p I
59
61 $3 65 67
Height in Inches
69
73
(b) B inomial D i st ribut i on
The binomial' distribution is represented by the expansion of the bin
(p + l) n . To understand the application of the binomial distribution to da
first necessary to make some study of probability. This subject will be tre
omiai.
ta,
ated
X is
later.
*Note: (p + q) n = p n + np n_1 q ■*■ n(nl) p^2 q 2
+ n(nl)(n'~2'.
1.2.3
■v
r??~
(c) Poisson Distribution
The Poisson distribution is biometrically unsymmetrical, i.e., it is extreme
ly skew. This type of distribution results from an attempt to represent the expan
sion of (p + q) n when "p" is extremely small. This type seems particularly applic
able to purity and germination counts in seed testing, as well as many other appli
cations.
• (3) Other Types of Distributions
Sometimes two or more factors influence the shape of a frequency distribution
so that it has two peaks. This would be a bimodal curve. When the data which pro
vide two unimodal frequency distributions with two substantially different means are
combined into one frequency distribution, the distribution that results may be bimo
dal due to the fact that nonhomogenous data overlap.* This happens occasionally in
genetic data.
IX. Some Constants used to Describe Distributions
There are several constants or statistics used to describe distributions. Those of
position or central tendency (mean, mode, median) have been discussed already. The
constants commonly used to measure dispersion of the variates are the standard devia
tion, quartile deviation, and the average deviation.
(a) Standard Deviation
The standard deviation of the sample (s') is most frequently used in statis
tical work to measure dispersion. It is sometimes called the standard error of a
single observation. The squared standard deviation (s')2 is the sum of the squares
of the deviations from the mean divided by the number. This is sometimes called
variance, or the second moment about the mean.
(s')^ (variance) = u 2 = S(x  x)^ _„_ (2)
N
where u 2 is the second moment.
The standard deviation (s') is the square root of the variance. The formula, for the
standard deviation may be expressed as follows:
= l ag + dg + d§ + . . .ag = /sa£ or st£ ll™™L™ (3)
where d is the deviation from the mean, e.g., &i » xj_  x.
The above formula gives the standard deviation of the. sample about its mean. When it
is desired to use this result as an estimate of the standard deviation of the popula
tion (s) about its mean (m), Nl should be used in the denominator instead of N.
This makes little difference in the result when the sample is large, but Nl should be
used when the sample is small, ile., when N is less than ">0 as an arbitrary rule.
As an example, the calculation of the standard deviation of a sample (s') can be
illustrated with the barley yields as grouped in VI (b) above. The deviations for
each class are taken from the actual means, i.e., 152.
*Hote: Pearson's generalized frequency curves or the GramCharlicr method of curve
fitting should be used for a finer method of analysis for such distributions.
1*1*
fx
31
1
31
50
1
50
69
k
276
38
12
1056
107
31
3317
126
69
869!+
3.1*5
80
11600
161+
97
15908
183
78
11+271+
202
21
1+21*2
221
1+
881+
2lf0
2
1+80
N
= 1+00
B(fx)
= 60812
X
— Ojla —
6
1)812 =
152 .0
s»
N
+00
= /Sf
12
/3910 : ^C
V N
J 1+00
jl
121
102
83
6k
1+5
26
7
12
31
50
69
88
fa
121
102
332
768
1395
179^
560
1164
21+18
1050
276
176
Sfd<
o
fM
' ll+61+l
101+01+
27556
U9132
62.775
1+661+1+
3920
13963
71+958
52500
1901*1+
15488
391050
 31 .27
Thus, 31*27 is the standard deviation (s') of this sample. However, the best esti
mate for the standard deviation for the population (cr)from which this sample was
drawn , would be:!
= /sfd 2
V NI
32^052 =31 . 51
Another formula for the calculation of the standard deviation of the sample (s f ) has
been recommended by J. Arthur Harris for machine calculation:
s ' =
bZ 1
57
W
This formula is essentially the same as the one given above except that the variates
themselves are used rather than their deviations from the mean. 2
The calculation of the standard deviation of the sample (s') by this formula is illus
trated with the barley yield data as follows:
Note: x The estimate (s) of the population standard deviation (a) may be computed from
the sample standard deviation (s 1 ):
Note
s = s
a = x  x
Sd d = Sx d
Sd 2 = Sx£
N N
IT
irti
,2
Nl
Sx c
TJ
Sxf
If/
d, d = (x  x)
^^2
x"
2x\x
x 2
_o
Sx + Hxr (remembering that Sx = Wi)
■ Sx^ + Zc. (remembering that
Sx = x . )
N
2xSx + Nx^ = Sxf
N N N
Therefore, /Sd
2
N
Sx 2
x 2
,2
V N
1 q Y '\2
if"
^
(1)
(2)
(3)
fx 2
X
f
fx
31
1
31
961
50
1
50
2500
69
k
276
190M+
x =
Sf x = 60 , 812
k i+oo
88
12
1056
9292&
107
31
3317
35^919
=
152.03
126
69
8694
1095W
Note :
Multiply column No. 1
11+5
80
11600
1682000
by column No. 3 to
16*
97
15908
2608912
obtain Sfx 2 <
183
78
1^274
26l2li+2
202
21
1+2U2
85688U
221
U
&m
19536' ] +
2^0
2
%Bo
115200
: S
(fx 2 )
N = i+00
s
(fx)
= 60812
9636298 =
8'
/Sfx2
W
(636298 (6o8l2\ 2
Uoo \ koo 1
31.27
= / 2J +, 090. 7^50  25,113.1209 = 31.2
(b) Coefficient of Variability (C.V.) is the standard deviation of the sample
(a 1 ) expressed in percentage of the mean. This gives a relative measure of disper
sion so that variation may be compared in features expressed in different units of
measurement. It would be often impossible to compare the variabilities of two ex
periments unless it was expressed in a common unit. The formula is as follows:
C. V. (Coefficient of Variability) = 100 s '  (5)
x
For the barley data, it is as follows:
C. V. = (31.27)(100) _ go 57
152.03
X. Sheppard's Correction for Grouped Data '".,,
An error is introduced by grouping variates into classes due to the fact that the
midpoint of the class is likely to deviate from the mean of the distribution by more
than the mean of the variates grouped in the class in question. This is particularly
true for the extreme classes. The majority of the variates in a class are grouped on
the side nearest the mean of the distribution. This error can be compensated for
mathematically by the use of Sheppard's correction. This correction is equal to l/l2
of the class interval (C), and is subtracted from the value of the squared standard
deviation (s 1 ) 2 as ordinarily obtained, i.e., (s*) 2  C 2 /l2. However, Sheppard's
Correction is applicable only to large samples where the. variables are continuous.
To calculate the standard deviation without Sheppard's Correction, is to assume that
the variates in each class are grouped with the highest frequency at the mean of the
class as shown in the diagram. To do this evidently leads to an error in that s'
will be computed larger than it actually is. Sheppard's Correction compensates for
this type of error which results from grouping data in a frequency distribution.
46
XI. Short Cut Methods for Computation of Statistl
OS
So far the statistics for simple frequency distributions have been calculated. Sever
al shortcut methods are used which greatly reduce the labor of computation. These
methods give the same results. Usually the computations are made from an arbitrary
origin or guess mean (v), with the guess mean corrected to give the true mean (x) of
the sample. The guess mean can be taken at any position.  Usually it is taken at
the middle of the range or at the lowest class.
The method of computation by use of an arbitrary origin, or guess mean, can be shown
with the barley yield data.
d
fd
fd £
31
50
69
88
107
126
145
164
183
202
221
240
1
1
4
12
31
69
80
97
78
21
k
2
1
2
3
4
5
6
7
8
9
10
11
i
8
36
124
345
480
679
624
189
4o
22
.0
1
16
108
496
1725
2880
4753
4992
1701
4oo
242
N = 400
Sfd = 2548 Sfd 2 = 17314
1
Note: It can be readily proved that a guess mean can be used provided a correction
is applied to obtain the true mean. Let x = the true mean, w = the guess
mean, C = a constant (class interval), d = the deviation from the guess mean,
and N = the number .
x = Cd + v
x = Sfx = Sf (Cd + w) =
W N
 Cd + w
C S(fd) + w s(f)
N N
since S (f) = N
47
Symbols: w = guess mean,' d or Sfd » correction to the guess at the mean, and
C = class interval.
S = Sfd = 2548 = 6.37
H 400
x = v (guess mean) + CcL .
= 31 + (19) (6.37) =31 + 121.03 = 132.03
8' = C /sfd 2 
o
cr or
C /sfd 2  /Sfd\ 2
V N \ H J
= 19 /i7^.
"V 4oo
= (19) (1.6456)
 /2^8\ 2
\ Uoo7
= 31.2664.
19 / 43.2850  40.5769
Sheppard's Correction:
s' (corrected) = /(s') 2  _P_ 2 = fan .5&lQ  ?6l
V 12 V 12
= /9T7.5878  30.0833 = / 9475045 = 30.7816
c. v. = loo s* = (30.7816) (100) = 3078.16 = 20.2471
x 152.03 152,03
The arbitrary origin in this case was taken at the first class. The calculation in
volves larger numbers than when taken near the center of the range, but all numbers
are positive. ■'■•
XII. General Applicability of Statistical Methods
Knowledge of the frequency distribution Isads to an elementary insight into the sta
tistical process. The methods of statistics must be applied with caution to experi
mental data. '•"■ ■; '•  .'. . ...,. •.'•••
(a) Mathematical Basis for Application
The methods of statistics comprise the application of the solutions affected
by the calculus of probability to precisely stated mathematical problems in the
attempt to answer questions connected with actual experiments. For the methods of
statistics to validly apply to the practical problems connected with experimental
work it is necessary that a high degree of correspondence exist between the realities
observed in phenomena and. the abstract but very definite concepts upon which the
mathematical solution of the problem is based. The one possible way to be certain of
a correspondence is to carry out repeated random experiments. Statistical methods
may be employed to answer questions and test hypotheses that concern phenomena ob
served in experimental work when this correspondence is satisfactory. The principal
cause of the misapplication of the statistical method is the fact that it is often
merely assumed that a correspondence exists between measurements and observations
concerned with phenomena that result from experiments and the abstract concepts of
the probability theory employed to produce the statistical method used in the inter
pretation of the experimental results. (See Keyman, 1937) • ■■
kd '■■'■ . ; .
(b) Value of the Statistical M ethod
There are many advantages attributed to the use of the statistical method.
(1) It provides a sound "basis for the formulation of experimental designs. Goulden
(1937) makes this statement: "The experiment that has "been correctly designed gives
maximum efficiency, an unprejudiced estimate of the errors of the experiment, and
yields results not only on the primary factors with which the experiment is concerned,
"but also on the important interrelations of these factors." (2) It tends to elimi
nate the personal equation, i.e., it does away with differences 'in personal inter
pretation. (3) The statistical method is useful in the reduction and condensation of
data. Fisher (193^) states that no human mind is able to grasp in its entirety the
meaning of any considerable quantity of numerical data. It allows one to express
relevant information by means of comparatively few numerical values, (k) It affords
a means to measure and evaluate chance errors. This is probably the outstanding con
tribution of statistics. (5) The statistical method affords one of the best measures
of concomitant variations, i.e., correlation. (6) It gives a quantitative measure
of variation, including chance variation. Statistics are widely used in genetics
for this purpose. . ..
(c) Reliability of the Stat istical Cjmst_arrt
The reliability that can be placed on statistical constants depends, in many
cases, on the type of data being analyzed. However, several factors contribute to
reliability, (l) Reliability depends on the accuracy of the measurements. (2) Quan
titative data are likely to be more accurately measured than qualitative data.
(3) Samples collected at random are usually more reliable than those selected by
other means, although samples by design in planned arrangements are very good. • (k) A
large sample is more likely to be representative than a small one. Arbitrarily, pop
ulations of less than 100 individuals or variates ordinarily are considered small
samples to which special precautions should be applied. (Fisher, 193^) • Conclusions
drawn from many of the older field experiments are questionable because there were
too many different kinds of treatments and too little replication or repetition.
Statistical methods have done much in recent years to increase the reliability of
field experiments. The difficulty of small samples has been alleviated in many in
stances by the calculation of a generalized standard error based on all the plots of
the experiment. Harris (1930) claims that many agronomic experiments can be organ
ized to "make possible the application of the powerful methods of 'biometric descrip
tion and analysis."
(d) Some Misconceptions o f th e Statistical Method ■„
There is little question about the value of the statistical method as such,
but much question as to its application. The statistical method cannot correct poor
technic or be applied indiscriminately. The standard error of a statistical constant
fails to measure the accuracy of an experiment unless, all errors ('personal equation)
have been eliminated except those due to chance. The statistical method may eliminate
some systematic errors, but to no great extent. An effective way tc eliminate syste
matic errors, or at least to discover them, is to repeat the experiment in a different
manner. Statistics may lend support, to a hypothesis but does not necessarily prove
it. Several 'years a.go, arguments on the use of statistical methods in agricultural
research were quite common. The mathematical foundations .of the statistical formulae
are now regarded as well established, but argument on the proper application of cer
tain statistical measures will continue much as it does in experimental technic
generally. Blind application of statistical procedures, as with any other technic,
is harmful. Common sense and good judgment. are vital, in all phases of experimental
work. Salmon (1929) points out that statistical treatment in Itself is seldom satis
factory because: (l) The observed result may not be due to the assigned cause.
(2) The laws of chance are often an unsatisfactory basis for action or for specific
advice. (3) Many experiments do not furnish results which readily lend themselves
.. .. .'. ■'„•', !»?
to statistical treatment because of bias, lack of randomness, or paucity of the
observations, (k) Most experiments furnish evidence supplementary to the main issue
which is of the greatest value for the arrival at a reasonable interpretation of the
results. This type of statement is answered by Goulden (1937) who "doubts very ser
iously the contention that all really worthwhile effects are obviously significant.
At any rate this is at best a dangerous concept as evidenced from scores of examples
in published papers where conclusions have been drawn that can be proved by the data
to have very little foundation Thus, the experimentalist who states
that his results are so obvious that they do not require tests of significance is
merely stating that in his experience with such experiments, differences as great as
those obtained are very unlikely to have arisen by chance variation. We have no
quarrel with this reasoning in that it is exactly the type of reasoning employed in
tests of significance. Our contention is merely that a determination of probability
based on a measure of variability furnished by the experiment itself is sound experi
mental logic and vastly superior to any method based on pure guesswork."
. References
1. Fisher, R. A. Statistical Methods for Research Workers Oliver and Boyd.
(5th edition) pp.. 17, and p. k9. 193^.
2. Goulden, C. E. Methods of Statistical Analysis. Burgess Publishing Co., pp. 18.
1937.
3. Harris, J. Arthur. Mathematics in the Service of Agronomy. Jo. Am. Soc. Agron.,
20:^31^. 1928.
h. Criticism of the Limitations of the Statistical Method. Jour.
Am. Soc. Agron., 22:263269. 1930. ...
5. Love, H. H. The Importance of the probable Error Concept in the Interpretation
of Experimental Results. Jo. Am. Soc. Agron., 15:217. 192J.
6. Neyman, J. Lectures and Conferences on Mathematical Statistics. Graduate School,
U.S.D.A. 1937.
7. Salmon, S. C. Why We Believe. Jo. Am. Soc. Agron., 21:854859. 1929.
8. . The Statistical Method:' A Reply. Jo. Am. Soc. Agron., 22:270271.
1930. ■  ••■
9. Tippett, Li H. C. The Methods of Statistics. Williams and Norgate. 2nd Ed.
pp. 19^2. 1937.
10. Treloar, A. E. An Outline, of Biometric Analysis. Burgess Publishing Co. pp.
420. 1935.
11. Weld, L. D. Theory of Errors and Least Squares. Macmillan pp. I30. 1916.
12. Yule, G. U. An Introduction to the Theory of Statistics. 9th Ed. pp. 211 213 .
1929.
Questions for Discussion
1. Explain why there is no such thing as exact measurement in quantitative data.
2. Distinguish between errors and blunders in measurements.
3. Define statistics. Why are some typical statistical constants?
4. In what branches of science have modern statistical methods been most extensively
used? Why?
5. Define these terms: variable, variate, frequency, population, and sample.
6. What is the mean? How does it differ from the socalled weighted mean?
7. What is a frequency distribution or frequency table?
3. What is a class interval? How would you determine it for an array of data?
9. What is the difference between a histogram and a frequency polygon?
10. Give 3 measures of central tendency and distinguish between them.
50
11. What is a normal curve? Skew curve? Bimodal curve?
12. Distinguish between the binomial, normal, and Poisson distributions.
13. Define the standard deviation of the sample. Population.
Ik. What is the best estimate of the standard deviation, of the population aa. obtained
from the sample?
15 . Prove that
'■d 2 = /5x2 /Sx\ 2
N V N \ N/
16. What is the coefficient of variability? When is it correctly used?
17. What is meant by an arbitrary origin? Where can it be taken? Why?
18. Explain Sheppard's Correction and the reason for its use.
19. What are some of the specific things that statistical methods are expected to do
when properly applied to data?
20. What are some of the difficulties likely to be encountered in applying statistical
methods to field experiments?
21. What factors contribute to the reliability of statistical constants?
22. Is the evidence afforded by statistical analysis of data negative or positive?
Explain.
23. Why have statistical methods been only partially used in agronomy? Name 3 men
who have advocated such methods in this field.
2h . What are the principal arguments of that school of opinion which favors (or in
sists) on the application of modern statistical methods to field experiments?
2p. What are the principal arguments of those who do not favor the use of such
methods?
26. What is generally indicated when "common sense" and interpretation based on sta
tistical methods do not agree'
Problems
In determining the moisture content of corn by the BrcwnDuvall moisture tester,
the common practice is to base the moisture percentage on the total or wet weight
(corn plus moisture) of the corn. The moisture content of hay, however, is often
expressed as a percentage of the dry weight of the hay.
(a) A variety of corn produced I.7.2 lbs. of shelled corn that contained 1^.0
per cent moisture on a 12 hill plot. The hills were 3x3 feet apart. Calculate
the yield of shelled corn in bushels per acre on a 15". 5 'P e ~f cent moisture basis.
(b) A twentieth acre plot of hay produced .120 pounds of field cured hay. Samples
taken when the hay was weighed showed that it contained 20 per cent moisture.
Express the true yield in tens per acre on a 15 per cent moisture basis.
Head counts were made on a number of fields in a township as follows:
FIELD NO. HEAD S COUNTED NO. HEADS SMUTT ED
1 50 " ^ ............ ...
2 1000 1
3 100 1
h 500 . 15
5 i+oo 20
6 300 h
7 1000 10
8 600 12
9 200 6
1Q 10000 __50
What percent smut may be expected in the wheat delivered to the elevator from
this township?
51
3. These data were taken from several fields to determine the probable losses from
smut for a community:
FIELD PCT. SMUTTED (X) SIZE OF FIELD (f)
(Uo.) (Heads) (Acres)
1 1.0 100
2 15.0 20
3 0.5 2:io
k 20.0 10
5 0.0 500
6 o.s 500
7 3.0 50
3 2.5 125
9 0.1 225
10 5.0 150
What percent smut may be expected?
k. Seme Iowa data were collected to determine the relation of certain ear characters
in corn. The yields from the very short ears, when used for seed, were as fol
lows:
Y ear No . ears Used Y leldBu. per acre
191? 2l 1*2 .70
1918 ; 2 .. 26.33
Determine the average yield for the very shoit ears for the 3year period.
5. The table that follows gives the heights of plants of buckwheat in a study of
variation at Cornell University. Plot the frequency curve on crosssection paper,
Height in
Centimeters 25 35 ^5 55 65 75 85 95 105 115 125 135 1^5 155
Number of
Plants 2 2 3 5 10 12 60 99 ikk 85 65 18 2 1
Total 508
Does this seem to approximate a normal curve? What can be said as to the posi
tion of the mean in a normal curve? The mode? The median? Is a normal curve
symmetrical? Is a symmetrical curve necessarily normal?
6. The table that follows gives the average yield of wheat per plant in certain
studies at Cornell University. Plot the frequency curve a3 in the previous
example .
IK
52
YIE LD PER PLANT N UMBER PLANTS
( grams )
0.5 57
1.5 59
2.5 88
5.5 kl
1+.5 ^5
5.5 29
6.5 26
7.5 5
8.5 8
9.5 6
10.5 ' 8
11.5 ' 5
12.5 5
135 1
1U.5 1
15.5 2
16.5 2
17.5 g_
Total 366
In what respect does it differ from, that of the previous example. What name is
given to frequency curves of this kind? Do the mean, median, and mode coincide
in this curve'?
7. The number of stalks were measured on two different kinds of Colsess barley
plants grown in 1930 at the Colorado Experiment Station. One kind was a normal
green (AcAc) and the other heterozygous for a lethal factor (Acac) . Plot the
frequency curves. Does the lethal seem to be detrimental to growth?
No. Stalks Heterozygous Plants Green Plants
per Plant (Frequency) (Frequency)
1 5 7
2 11+ 9
3 51 28
k 62 33
5 63 31
6 1+1 19
7 21 12
8 12 k
9 5 1
10 1
11 1
12
13 1
Totals 27"S~ lkj
(Note: Calculate the frequencies of the green plants on a basis of N = 276 in
order to make the two sets of data readily comparable.)
Some data were collected by Emerson (1913) for the study of size inheritance in
corn. Classify the data for hybrid 60 x r jk . Prepare a frequency table for
these data and calculate the mean of the sample using a guess mean. Continue and
find the standard deviation (s') ana the coefficient of variability. The measure
ments are given as lengths of ears in centimeters:
53
Hybrid 60 x jk
15 13 10 12 13 10 13 15' 11 10
10 13 15 12 13 Ik lk lk 11 10
13 12 11 12 11 12 10 13 Ik 12
11 11 Ik 10 9 10 11 13 13 1^
12 11 10 Ik 11 13 12 13 13 10
11 12 12 11 13 12 10 13 12 10
11 13 Ik 13 12 15 1*+ 12 13
9. Calculate the standard deviations (s v ) for height of plants in problem p, using
(a) deviations from a true mean and (h) deviations from a guess mean.
10. Some 1930 data on black hulless "barley plants were compiled by the Colorado
Experiment Station to determine the variation in number of kernels per plant.
The data are grouped in classes, (a) List the class boundaries, and calculate
the mean, standard deviation, and coefficient of variability, (b) Apply
Sheppard's correction to the standard deviation, (c) Is the number of group
classes sufficient according to Yule's formula? Calculate.
x (class center) 15 ^5 75 105 135 165 195 225 255 285 315
f (frequency) 2 12 11 26 38 26 18 13 lj 3 1 = I63
Note that the origin is taken at the class center below 15 .
CHAPTER VI
TESTS OF SIGNIFICANCE
I. Sta tistics as a Basi s for G eneralizatio n
So far, the discussion has dealt with, a sample and its statistical description. The
investigator may desire to apply the information collected from the samples to de
scribe the general population,. Before he can do that, he must take into considera
tion the chance or random errors introduced in the actual taking of the sample.
Chance errors result from the operation of a great many factors, none of which is
dominant, and all of which are relatively similar, equal, and independent. When only
chance errors operate, the data are said to be random and follow the law of gr'eat
numbers.
Two kinds of error exist, chance and systematic. Errors due to chance may not "be
entirely eliminated but can be submitted to' mathematical treatment. Systematic
errors can be largely eliminated when an experiment is properly planned.
II. Theory of P robability
In the analysis of chalice errors, it is necessary to introduce some of the fundamen
tal concepts of mathematical probability.
(a) Single Probabilitie s
The probability of the occurrence of an event can be defined from two view
points .
(1) Mathematical Probability: The mathematical or a pri ori probability of an event
is the ratio of the number of ways the event may occur to the total number of ways
it may either occur or fail to occur, assuming all such ways are equally likely.
Thus, the probability of drawing any individual card from an ordinary deck is l/52,
while that of drawing any card of a given suit is 13/i>2 or l/U. Probabilities are
sometimes stated in terms of odds, e.g.; suppose the probability of the occurrence
of an event is l/2p. The odds are 1:24 in favor of its occurrence, or 24:1 against
its occurrence. To be more explicit, the occurrence of the event is expected just
once in 25 trials.
(2) Statistical Probability: Suppose an experiment is repeated a great number of
times. When it terminates in a particular manner a certain number of times, the
ratio of this latter number to the total number of trials defines an estimate of the
probability of the particular termination. Suppose N and N represent the number of
successes and the number of trials (both successes and failures), respectively, then
Limit _N_}_ will be defined as the probability
of a success, Thus this probability can be approached but never attained in practi
cal work with infinite populations. The permanency of the value N /N for N large is
the law of great numbers. This permanency results from randomness in the experimen
tal trials and is the necessary property that statistical data must possess to admit
valid treatment by mathematics. As an illustration. of statistical probability, in a
frequency distribution, any particular class frequency divided by the total number of
observations in the distribution gives an estimate of the probability that any indi
vidual observation made at random will fall in that particular class.
It is evident from either definition that the probability of the occurrence of an
event may vary between zero (0), i.e., certainty that the event will not happen, and
55
one (1), i.e., certainty that the event will happen.
(b) Several 'Probabilities
When several probabilities are to be dealt with simultaneously, it becomes
necessary to consider two fundamental theorems.
(1) Theorem IY When a number of mutually exclusive events have certain probabilities
of occurrence, the probability of occurrence of some one or other of these events,
is the sum' of their individual probabilities. For example, tho "probability that an
observation in the barley yield data (Chapter 3, pages 39 and ^0) will fall in class
x = 88 is 12 /lj00, while the probability that one will fall in class x = 107 is 3lA 00 »
The probability that an observation will fall in either class 88 or 107 is 12/U00 ■*■
3lA00 = U3/U0O, i.e., P = 0.11.
(2) Theorem II: When a number of independent events have certain probabilities of
occurrence, the probability of all occurring together is the product of thoir individ
ual probabilities. In the above example, the probability that the first and second
observations will fall in classes x = 88 and x = 107, respectively is 12/kOO times
31A00 = 372/160,000, i.e., P = 0.0023.
A  Large Sample Theory
III. Probability and the Normal Curve
Statistical data that possess the property of randomness often are distributed in a
manner closely expressed by a normal distribution. Many of the sample statistics of
large samples can be mathematically proved to have distributions extremely close to
normal. Therefore, the application of probability to the normal curve is important
in practical work. The area below the curve is taken as one unit . Hence, the area
between any two ordinates may be considered as the probability that an individual
observation will fall within the range defined by the two ordinates. Wow, by theorem
I, tho probability that an individual observation will fall within any range, is the
sum of the probabilities that it will fall in all subdivisions of that range.
Thus, for characters which are distributed normally, it is possible to estimate the
probabilities of their occurrence in any given range. This is done by finding .the
areas beneath the normal probability curve t,hat correspond to the given range. Math
ematical tables of such areas, called probability integral tables, have been con
structed. (See Table I in appendix).
Some of the most important probabilities and ranges are given below with the aid of
a figure.
t =
3a
(* = 
2cx lo^P.Ey ^P..4lcr +'dcr
0.67^5 and < 0.67^3, respectively)
56
In the case of a normally distributed variable, it is clear that: the probability that
an individual observation will fall within a range of a on either side of the mean
(x) is approximately 0.68; within a range of 2 er it is approximately 0.95; while with
in a range of 3 cr it is 0.997 Thus, the probability that the observation will dif
fer from x by at least 2 <r is 1.00 . 0.95 or about 0.05. In other words, the chances
that an individual will fall outside a range of 2 o  are approximately 5 : 95 or 1:19
This means that such a situation may be expected about once in 20 times due to chance
alone. In like manner, the probability that the observation will differ from x by at
least 5 cr is 1.000  0.997 or 0.003. Such a result, then, may be expected to happen
only once in 333 times. Therefore, when an observation differs from the mean by
"too much" there arises the important question as to whether or not this abnormal re
sult might not be due to some special cause acting in the case of this individual.
When some special affecting condition is known to exist, common sense leads one to
the conclusion that the extreme abnormality of the observation is more likely due to
the affecting condition than to be expected on the basis of probability.
TV. levels of Significance
What constitutes an abnormality which is "too much" is a matter of arbitrary decision.
Common usage in this country considers an abnormality of twice the standard deviation
(standard error in this sense) as being sufficient to warrant the statement that the
abnormality of difference from the mean is a re al or significant differ ence. 1 This
does not mean that an individual observation taker, at random and showing a signifi~
cant difference does not belong to the general population. However, in such a case
one would inquire as to whether the individual case in question was of a special
nature, either inherently or by reason of treatment. Should such a condition be sub
stantiated, it is quite proper to attribute the abnormality to special cause or con
dition and not to chance.
Some workers in the field of statistics use a difference of 3 cr as a criterion for a
significant difference, This allows the worker to place more confidence in a con
clusion derived from a "significant" observation, but this advantage is overshadowed
by a possible tremendous loss cf information due to the imposition of a too stringent
criterion.
V. Different Kinds of P ro bability Tables '
There are two kinds of probability tables, viz., oneway and twoway tables. The use
of a particular one depends upon the nature of the statistical hypothesis to be test
ed. The results obtained in one can be readily explained in terms of the other (See
Livermore, 193 1 ' ) •
(a) OneWay Tables
The principal oneway table for normal curve areas is that devised by
Sheppard and published by Karl Pearson (191*4) as Table II. Suppose an ordinate is
erected at a distance on the positive side of the mean, exactly twice the standard
deviation (.cr) , Thus t or d/a = 2. From Table I (appendix), it is found that the
area (A) that corresponds to t (or d/a) = 2 is 0.9772, or the area defined by the
interval from minus infinity to the assigned value of t (d/a). Thus, with the total
area beneath the curve considered as 1.0, the area' to the left of the ordinate is
0.9772 while that to the right is 1.0000  O.0772 = 0.0228. Thus, P = 0.0228
(about l/kk) is the probability that a value taken at random, will exceed the mean
( in one direct ion only) by an amount equal to 2 or more times the standard deviation
(a).
lThis approximates 3 times the probable error.
57
Sometimes probabilities are expressed as odds:
Area inside the ordinates divided "by area outside the or di nates is equal to the odds
against the occurrence of a deviation as great or greater than the designed one due
to chance alone. In the above example, 0. 9772/0.022 8 = kjil (approximately)
In this case the odds are k$:l that a value will not exceed the mean to the extent
of two or more times the standard deviation due to chance alone. Table I (appendix)
is a one way table.
(b) Twoway Tables
Suppose one inquires as to the probability of selecting a variate at random
so that it shall fall outside the limits of plus or minus twice the standard devia
tion. Two ordinates are erected, one at t or d/a = 2 and one at t = + 2. The
problem is to find the area in both tails of the curve. This will be (1.0000 
0.9772) times 2 = 0.0k^6, or double that in the oneway table. This means that the
probability that a single variate selected at random will deviate by an amount equal
to or, greater than + 2a is O.O^b, or approximately l/22.
The values on a twoway basis can be expressed as odds as follows:
0.954^/0.0^5^ = 21:1. Thus, the odds are 21:1 against the occurrence of a deviation
as' great or greater than the designated one (plus or minus twice the standard devia
tion) due to chance alone. A typical twoway table for large samples is Table IV
given by Davenport (193&).
In summary, it should be clear that the oneway interpretation or the use of
a oneway table gives the probability or odds that an obtained value shows a certain
discrepancy from the mean in a stated direction whereas the twoway interpretation
does not state the direction which the discrepancy must take in a statement of proba
bility or odds .
( c ) Transformation of Values  ■
The probability values obtained in one type of table. can be readily trans
formed into terms of the other to. meet the experimental argument at hand. Probabili
ty values in a oneway table can.be doubled to give the results obtained from a two
way table, and vice versa.
The transformation of odds is as follows:
Odds in twoway table = odds in oneway table  1
2
Odds in oneway table = I (odds in twoway table) (2) 4 1. I
' VI * Standard Errors of Sta tistical Constants
Each statistical constant or estimate has its own standard error. The standard
error of a statistic derived from a sample is the standard deviation of the distri
bution of that statistic thought of as resulting from many samples. The distribu
tions of many statistics are nearly normal, particularly when the basic sample is
large.
(a) Standard E rror of a Single Observat ion
The "best" 1  estimate of the standard deviation of a single observation (cr)
is the standard error (s) derived from the sample. Some data on the total weight of
^Note: The best unbiased estimate is simply called •"best''. See more advanced treat
ments of mathematical statistics.
58
grain in grams for non competitive Colsess barley plants as follows:
Class Center 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33
Frequency 3 11 21 35 kj 55 7I 52 k'J 35 21 10 9 11 5 1 1
N = ta x = 13.8 Sfd 2 = l^lG.bh
s = standard error of a single observation = /Sf i c (l)
■V Wl
In the above example, it would be calculated as follows :
= / ^ j j _~y  6.003 grams
This value, s = 6.003 grams, is the standard error of a single Variate in this sam
ple. For instance, the value of the mean, 13.8'+ 2 (6,003) indicates that the odds
are 21:1 that a single individual taken at random will not deviate from the mean by
more man 2 a in either direction, where the normal distribution of the population
affording the sample data is assumed. NV
(b) S tandar d Error of t he Mean
Suppose a second sample were taken. One could hardly expect to get exactly
the same result for the mean (x) as in the sample in question. Thus, the mean (x)
obtained from a single sample is merely an estimate of the true mean (m) of the whole
population. The latter is unknown and necessarily must remain so. In case it were
possible and practical to take and analyze a greater number of samples, finding the
mean (x) for each, one would expect the mean of all the sample means to be very close,
indeed, to the mean of the population (m) . Since this is not feasible, one can only
ask how good an estimate of the population mean (m) is the mean (x) computed from a
single sample. The answer to this question can only be given in terms of probabili
ties. It can be shown mathematically that the mean computed from a large number of
large samples are distributed nearly normally with standard deviation, ox, , which
is theoretically equal to the ratio of the standard deviation of the population to
•JW, the number of observations that make up the sample*? However, the standard devia
tion of the population (o) is unknown and in its stead its estimate (s) derived from
the sample is used. Therefore the standard deviation of the hypothetical distribu
tion of means of a large number of samples will be estimated as follows:
c = standard error of the mean ~ s • (2)
x
/¥
The greater the number of observations in the sample, the smaller will be the stand
ard errors of the various statistical, constants. Hence, the statistical constants
derived from a large sample are more likely to represent the true constants of the
general population than those derived from a small sample. When the sample is small,
the argument is the same except that the distribution for x deviates from normality
and needs special interpretation.
V^It should be noted that s = /Sfd , which best estimates the standard deviation
V¥i ' _.
of the population, closely approximates s' = /Sfd which is the standard deviation
V N
of the sample, and is the estimate of c  given by the maximum likelihood principle.
villi g is sometimes expressed as ''S »E . of the mean."
59
In the above example the standard error of the mean {°%) is:
o~x = _S_  6.005 = O.296 grams.
/n Jku
The mean is I3.8 + 0.296 grams. Therefore, the odds are 21:1 that x (I3.8 grams)
dees not differ from the unknown true mean (m) of the general population by more
than 2oj, or 2(0.296) a 0.592 grams.
(c) Standard Error of the Standard Dev i atio n
Next, it is desired to discover how reliably the standard deviation of a
single sample (s 1 ) estimates the unknown standard deviation of the population (cr).
Mathematically, it has been found that the best estimate of a hypothetical distribu
tion of standard deviations derived from a great number of samples is as follows:
= standard error of standard deviation = _j3_ (approximately) (3)
o" f~ —
y2N
From the example used above,
a cr = zM= " 6  QQ 3  b 0.209 grams
Therefore, the odds are 20:1 that s = 6. 003 grams does not differ from the unknown
true standard deviation of the general population (c*) by more than 2 a or 2 ( 0.209)
= 0.4l8 grams.
(d) Standard Error of the Coefficient of Variability
By use of the same argument, the standard error of the coefficient of varia
bility is:
°" = C. V. ^1+2 / C.VA 2 2 when C. V. is  (k)
C ' V * /~2N ■ \ 100 ' J greater than 10
°>. «• = c ' V » vnen c  v  is le s s than 10  ( F
c .v . . . 
/2N
In the example used.,
a = *35
c.v.
M2fi4.nl
1 +2 fk^2
100
2 = I.78I
A table has been worked out by Brown (193*0 to shorten the computation necessary to
secure the standard error of the coefficient of variability when C.V. is greater than
10.
(e) Standard Error of an Average of Averages
The standard error of an average of averages is given by the formula:
°a » 1
/*J"% * °% ^ *\ ■ •••< 6 >
where N equals the number of separate means and a £ a 5^, etc., represent their
separate standard errors. c 
(f ) Standard Error of a Difference
Suppose two samples are measured with respect to a common character. From
the data, let two similar statistical constants be compared, e.g., the two means or
6o
the two standard deviations. The question arises as to whether the two constants,
differ significantly. Its answer depends upon the standard, error of the difference
which is as follows:
1
°a = V (j2 l + °" 2 2  / a ~i * eC 2 (7)
where sj and So are the standard errors of the two like statistical constants derived
from the two samples. Where a significant difference results In the case of two sam
ples drawn from the same population, it would Indicate probable improper sampling
technique leading to lack of randomness.' The principal' use of this method lies In
its test as to whether or not a fa/ctor known to exist in the case of one sample, and
not in the other, is really a causal factor to which an abnormal difference can he
attributed, e.g., the difference between two yields in a yield triad.
For example, suppose it Is desired to determine the standard errors of the difference
of the mean yields of Kanred and Turkey wheats, and also for Manchuria and Minnesota
hk^ barleys.
(1) Wheat Variety Yield (Bu.J (2) Barley V ariet y Yiel d (Bu .)
Kanred 25 + 0.7 Manchuria 38*9 + 0*9
Turkey 2k + 0.6 Minnesota kk5 1+8.5 + 12
Difference 1 4 0.92 Difference 9.6 + 1.5
l7 d = 7(°T) 2 + (0.6) 2 = 0.92 o rl = VTo.9) 2 + (I?) 2  1..5
VII. Significant D ifferences
After a difference Is obtained for two statistical constants, as in the above example,
it is desirable to test this difference for statistical significance. An investigator
may arbitrarily choose whatever level of significance he desires, but should state the
level chosen. He must use care In attributing differences to causal factors when the
differences approach the level of significance that he has chosen. To determine the
significance of two statistical constants, their difference divided by the standard
error of the difference (ctcl)Is commonly employed. For example, in the case of Kan
red and Turkey wheats cited above,
t = d_ = JL_ = 1.09
og O.92
When the level of significance la taken as d/c&  2, this difference is not signifi
cant. On the basis of probability, the odd.s are a little more than 2:1 that this
difference is a real or significant difference. Hence, it may be ascribed to chance,
or to put it in another way, one would not claim superiority for Kanred because the
probability is too large that such a statement is incorrect.
In a comparison of Manchuria and Minnesota hk^ barley,
t ■ =jl = _9._6 * 6 .'!
Hlhen Oj = op, a d = qJ2, ....
^Relation (7) holds strictly only where the two variable statistics are normally dis
tributed and derived from uncorrelated data.
61
Since t = d/oft is far greater than 2, the difference in yield between the two varie
ties is said to be real and not due to chance . In this case the claim is made that
Minnesota Mt5 is superior to Manchuria in yield ability. The probability of the
incorrectness of this statement is insignificantly small. It would be a miracle if
this claim were really incorrect.
VIII. Probable Errors .
The quantity O.6745o~, which gives the range that contains half the observations, is
termed the probable error. For example, an average yield of 15.0 +1.5 bushels would
mean that the chances are 50:50 that the true value of the average for an infinite
population lies between 13.5 and I6.5 bushels. It also indicates that the chances
are even that it may lie outside this range.
(a) Use of Probable Errors
Historically, the probable error was used before the standard error. It is
still widely used in this country in the statistical treatment of biological data,
but the tendency is to use the standard error and think in terms of it. It is gen
erally felt that "the probable error is an unmitigated nuisance," and has nothing to
recommend except its previous usage.
(b) Formulae for Probable Error s
The probable error is approximately twothirds of the standard error. It can
be obtained when each standard error value is multiplied by 0.67^5. These formulae
may be briefly summarized:
(1) P.E. single determination = + O.67U5
(8)
(2) P.E. = 0.67^5 8' or + 0. 67*15 s
(9)
/ N  1
/r
(3) P.E._ = 0.67^5 s' (10)
cr
J 2N
(h) P.E. = + Q.67^5 C .V.
c.v.  L_ y
V2N
1+2 jC.V.) 2
1 100
(11)
where C.V. is greater than 10.
(5) P.E. C#V> = + 0.6745 C.V. (12)
■J 2N
where C.V, is less than 10.
(c) Levels of Significance for Proba ble Errors
The level for significance for the probable error is commonly taken as
D./P.E.^ = 3' This is equivalent to odds of about 22:1. Some workers use 32 times
the probable error, for which the odds are approximately 30:1. A table of odds for
probable errors is given by Hayes and Garber (1927) in "Breeding Crop Plants."
(d) Relation of Standard Errors to Probable E r rors
Based on the normal curve the quart ile lines, Q2 and Qu,
error of a single variate, or Q = O.67J+5 cr.
give the probable
62
The intervals
£ +1 o*
x + 2 a
X + p CT
include
>8.3 ? of variates
9 E >.5
QQ.7 " "
997
x + l P.E.
x + 2 P.E.
i + 3 P.E.
Include
50.0 'jo of var later,
95.7 " "
B  Special Case of Small Samples
IX . Use of Small Samples in Biolog ical Research
The methods heretofore explained relate to the determination of significance based on
the normal distribution for large samples, but it is not always possible to obtain
large samples. This is often the case in agricultural or biological experiments.
When the investigator can be certain that the populations which afford, small samples
approximate the normal distribution in form., he may feel that the interpretation of
the statistical analysis was valid. Therefore, the materia], that follows is given on
the basis of small populations whose distributions approach that of the normal curve.
Statistical treatment of small samples, from populations far from normal in distri
bution, may probably be inadequate. Too often it may lead to Incorrect conclusions.
Statistical analysis of a single sample with less than 20 cases is hazardous. In
samples of 20 to 100 cases, the hear normality of the underlying population should
be known. This places a severe limitation "on the use of small samples, but fortunate
ly in agricultural and biological experiments, most of the populations with which the
experimenter deals, are near normal. The importance of the small sample, together
with its statistical treatment, has been discussed by Fisher (193 1 !)
X. Degrees of Freedom
The reliability of a statistic (estimate of a population parameter) will obviously
depend upon the number of variates in the sample. This dependence is also affected
by the number of restrictions placed on the aggregate observations in the determina
tion of an estimate of a population parameter. The total number of observations
diminished by the number of restrictions which they in aggregate must submit to has
been termed "degrees of freedom" by Fisher (193*0 •
It has been stated that the best estimate of the variance of a population a3 derived
from the sample is as fellows:
S(x  x)
2
(13)
In this case, the number of individual observations (N) Is diminished by one to give
the degrees of freedom. The number of statistical constants of the sample which are
directly used in the computation arc subtracted. The mean or total fixes one value
in the above formula, so that only IJ1 observations are free to vary. This Is of
little importance when a large sample Is analyzed, but very important in small sam
ples.
XI . Probability Determinations with S mall Samples
The distribution of x I _§_ is not sufficiently close to normal for small samples. The
nature of the distribution of x/s' was found by "Student" in I90S. He prep ared a
series of tables based on the distribution of s' (whore s' = V S ( x  S^/N) which he
designated as "Z" . He showed that the "Z" distribution, now more commonly called
63
Student's distribution, was the same as the Pearson Type III curve. More recently,
he has prepared tables for the distribution of "t" which is designated as x/oj or
x >/n7 8 by Fisher (193*0 • For a given value of "t" that corresponds to a given number
of degrees of freedom one can read the probability in an analogous manner to the way
the tables of areas of the normal curve are used.
The "t" table devised by Fisher (193*0 is a twoway table. A probability of 0.05 is
Fisher's 5 per cent point for which the odds are 19:1; a probability of 0.01 is the
one per cent point for which the odds are 99:1 . In addition several oneway tables
are in use. These are as fellows: (1) Student's "t' ! table, (2) Livermore's modifi
cation of Student's "t", (3) Student's "Z", and (k) Love's modification of "Z". For
example, suppose t = U.60^ for k degrees of freedom. The probability as found in a
oneway table is equal to 0.995. The calculated odds would be 199:1. They are cal
culated as follows: 1P = 1.000  0.995 * 0QP5. 5/1000 = l/200. P = 1/200 is
equivalent to odds of 199:1
XII. Significance of Means
When d is the difference between the mean of the sample and any value (m 1 ) assumed to
be the mean (m) of the population, it has been stated that the difference, d = x  m',
is significant when d/a x exceeds 2. When this occurs, the hypothesis (m .= m') is
rejected. This procedure holds when d/o g is nearly normally distributed as in large
samples. As this distribution is not close to normal for small samples, the "t" table
should be used in such cases. When the 5 per cent point is used as the level of sig
nificance, a value of t = d /<j£ that corresponds to P = 0.05 is considered as signifi
cant. In this test for the significance of the mean one determines the probability
of drawing a sample with a msan equal to x from a population whose true mean (m) is
assumed to be some particular value (m 1 ).
XIII. Means of Two Indepen dent Sa mp les
One of the most important problems in statistics is to test the significance of a
difference between two means, i.e., 2,  x 2 = d. Previously, it has been stated
that the s tandard error of the difference of the means of two samples is o^ =
rX, * °xo* Should there be any reason to suspect that the standard deviations of
the two underlying populations are different, one should form t/og_ with o^ as given
h^re.
(a) Samples with Different Numbers of Obs ervations
When it can be assumed that the standard deviations of the populations are
the same, or that the samples have been drawn from the same population, then the
best estimate (s) of the population standard deviation (cr) is:
s = / D ( Xt  x
x x ) 2 * S(x 2  *2)2   (13+)
(1^  1) + (N 2  1)
Here N^ and N 2 are the numbers of observations in the two samples while the denomina
tor evidently denotes the degrees of freedom. This method to determine s as an esti
mate of o* is particularly important in the case of small samples.
6k
The "t" value, equivalent to d/s^ is calculated as follows
t \/ = xi  xo r~ ~ —
i :i_ ; xm i i^o
V NfTSa (15)
(b) Samples with Same Number of Observations
The above formulae are simplified when the number of observations are the
same in each sample, i.e. N^ = No.
The standard error (single observation) is as follows:
/■
3 (xj  x 1 ) 2 + s(x ?  x 2 ) 2  _ .___.._ _ _ _ (16)
V 2 (If  1)
The value of "t" is as follows:
i ^2  K I ,
Some data presented by Imiuer (19J6) may be used to illustrate. the computation. Sin
gle plots of Velvet and Glabron barley were grown side by side in single plots on 12
different farms. The yields in bushels per acre are given below:
Farm No. Glabron (xj ) Velvet (xo) Sum
1 '+9
2 1+7
3 39
k 37
5 hG
6 52
7 51 . •
8 . 57
9 *+?
10 1+5
11 1*8
12 6'+ .
1*2
91
J 47
oil
38
77
^2
69
111
87
lid
93
] '5
06
5 ...
113
1*2
87
39
34
^7
95
59
105
S (x 1 ) = 58O S(x 2 ) = 509 IO69
*1 r " ^8.3333 x 2 = 1*2 "Ja67 x ^ ! ^5.3750
S( Xl 2 )  28,620 S(Xg) = 21,979 100,269
(Sx r )2 = 28,033.31 '" (Sx 2 ) 2 =; 21,590.10
V This can bo readily proved as follows
r'.T
X]_  x 2 = x.^  x 2 = xi  X2 = xj  Xg I N^Ng
2 • 2 2 2 ; s I N
  So „ WoS,  IT, So BL No  Nn , I 1X 1
% W 2 I *1* 2 /X W 2
where "3" is an estimate derived ~o'j pooling the two samples, based on the hypothesis
that the two populations have a common standard deviation (o).
6 5
The computations are as follows:
. [S ( Xl 2 )  (Sx^/Nj + [s (x 2 2)  (SX2) 2 /N 2 ]
2 (Nl)
= (28,620.00  28,055.51) + (21,979.00  21,590.10)
22
= 586.69 + 588.90 = 97559 ■ ^.5^50
22 22
s = Jkk.3k50 = 6.6592
t = x x  x 2 [W = ^8.55  te.te /if = 2#1?69
V2 6.6592 V2
The "t" table is entered for t = 2.1769 for 2 (nl) = 22 degrees of freedom. P lies
"between 0.05 and 0.02. It may. he concluded that the odds are in excess of 19:1; that
the difference "between the mean yield of these two varieties is not due to chance.
XIV. Means of Paired Samples
In this case, the variahles are paired, i.e., each value of x^ is associated in some
logical way with a corresponding value of x 2 . As a result, there will he the same
number of variates in the two samples. When there are II pairs there will he Nl
degrees of freedom available for the comparison. This is widely known as Student's
Pairing Method.
(a) Student ' s Pairing Method
This method is devised to compare two results on a probability basis. Ii, is
used primarily for small samples it not being necessary to assume a normal population.
Partial mathematical proof of the method was first published by Student (V.S .Gossett)
in 1908. Differences between paired values are dealt with directly, with the result
that the correlation between paired values is taken into account. The method was.
brought to the attention of American agronomists in 1925 by Love, et al. (1925* 19 2) +).
The variance (s 2 ) and "t" values are calculated as follows:
  (18)
s2 = variance = S(d 2 )
 (Sd)2/N
/ . N
 1
% a / HF
 (19)
Here "t" is used to test an obtained, value, d, in accordance with the hypothesis that
the mean of the population of differences is zero. A significant result would mean
the rejection of the hypothesis and would warrant a statement that the mean of one of
the basic populations exceeded that of the other.
(b) Method of Computation
The method of computation can be illustrated from the Glabron vs Velvet barley
yields mentioned above. The computation follows:
66
Farm No. Glabron (xi ) Velvet (xp) ■. d (Velvet 'from
Glabron)
1 k9 k2
2 »47 lt7
3 3^ 33
k • 37 32
5 h6 hi
6 52 H
7 51 *'.' ^5
8 57 56
9 ^5 ! +2
10 1+5 59
11 >+8 k'j
12 64
5V
bum t>
(x x ) = 530 S(xp) = 509
Q
7
1
5
R
s
11
6
1
3
1
25
s(a) =
71
a 
5.9167
Mean % = ! R>.3333 3Eg > } 42.>+l67
(d 2 ) = 929 (Sdj 2 /n = ^20.0857
s2 = S(d 2 )  (Sd) 2 /p * 929.OOOO  U20.0857 = V6. 263+9
.N  1 11
t = C W /jE" = 5.9167 / lh' 6.26^ ,= 5.9167. / /5T8555 = 3.0133
The value of "t" ia then looked up in the t table (Fisher, 1930 ) for 11 degrees of
freedom (lll paired values) where it is found that the observed value lies "between
1 = 0.02 and P = 0.01.
The "Z" table devised by Student is sometimes , .!.sed. He designed "Z" as the ratio
of the mean difference to the standard deviation of the mean difference, i.e.,
Z = Jc where s ' = /S_( xx) 2
B 1 V dJ '
Student (1926) calls attention to the fact that the "Z" table should be enter 3d with
Ni degrees of freedom. As mentioned previously, his "Z" table is a oneway table.
The Zvalue can be transformed to "t" as follows:
( c ) Application of the Pairing Method
The application of "this method is highly desirable for making comparisons
between pairs of varieties or treatments when the scope of the experiment is limited
to a few pairs of observations. It is useful for simple tests such as nested vs.
untreated where only two or three things are being compared. In plot work, the
method can only be used to remove soil heterogeneity where the plots are physically
paired, i.e., adjacent .
References'
1. Brown, Hubert M. Tables for Calculating the Standard Error and the Probable
Error of the Coefficient of Variability. Jour. Am. See. Agron., 26:6569.
J93 1 ^ •
2. Davenport., 0. B., and Ekas, M. P. Statistical Methods in Biology, Medicine, and
Psychology. John Wiley and Sons . pp. 35 ) +0, and pp. 166172. 1936.
3. Fisher, R. A. Statistical Methods for F'" search Workers (5th edition). Oliver
and Boyd, pp. 112125. 193<+.
6 ?
k. Goulden, C. H. Methods of Statistical Analysis. Burgess Publ. Co., pp. 9H;
and 2026. 1937.
5. Hayes, H. K., and Garber, R. J. Breeding Crop Plants. McGrawHill, p. h2 &
8692. 1927.
6. Immer, F. R. Manual of Applied Statistics. University of Minnesota. 193^
7. Livermore, J. R. The Interrelations of Various Probability Tables and a Modifi
cation of Student's Probability Table for the Argument "t". Jour. Am. Soc.
Agron., 26:665673. I93U.
8. Love, E. H. The Importance of the Probable Error Concept in the Interpretation
of Experimental Results. Jour. Am. Soc. Agron., 15:217225. 1923.
9. Love, H. H., and Bruno on, A. M. Student's Method. Jour. Am. Soc. Agron., Io:60.
192^ .
10. Love, H. H. A Modification of Student's Table for Use in Interpreting Experimen
tal Results. Jour. Am. Soc. Agron., 16:6873. 192^+ .
11. Pearson, Karl. Tables for Statisticians and Biometricians. Part I. Cambridge
U. Pres3. pp. 28 (Table II). IQlU.
12. Student. The, Probable Error of the Mean. Biometrika 6:125. I908.
13. • New Tables for Testing the Significance of Observations. Metron,
5:1821. 1925.
Ik. Mathematics and Agronomy . Jour. Am. Soc. Agron., 18:703720. 1926.
15. Tippett, L. H. C. The Methods of Statistics. Williams and Norgate. (2nd edi
tion), pp. 110121. 1937.
Questions for Discussion
1. What is the basis for using statistics for generalization?
2. Distinguish between a prio ri and statistical probability.
3. Give two basic theorems where several probabilities are involved.
k. What is the geometrical significance of the standard error? Its significan.ee in
practical problems?
5. Why is a difference said to be statistically significant when it is two or more
times the standard error?
6. Is it correct to say that standard error is a measure of experimental error?
Explain.
7. What is the difference between a oneway and twoway table in the calculation of
probability? Interpret probabilities calculated from each kind of a table.
8. How do odds differ in on^way and twoway tables?
9. How can odds be transferred from a oneway 'to a twoway basis? Explain the dif
ference in interpretation.
10. Explain the difference between the standard error of a single observation and
the standard error of the mean. Give the formula for each.
11. What is the formula for the standard error of an average of an average? Standard
error of a difference?
12. What is the relation of the standard error to the probable error? Why do most
statisticians prefer to use standard error?
13. Why are special methods used for small samples?
lb. What is meant by "degrees of freedom"?
15. Who was "Student"? What were some of his contributions to statistics?
16. What is the meaning of Fisher's "t"?
17. What was Student's "Z"? How can it be transformed to "t"?
18. How is the standard error calculated for the means of two independent small
samples drawn from populations with equal standard deviations?
19. What is Student's pairing method? How does it differ from other methods of cal
culating standard errors?
20. Under what conditions can Student's pairing method be used? What are some of its
limitations?
68
PROBLEMS
1. (a) If the mean of a population is 21.65, and o~ ■ 3'21\> determine the probabili
ty that a variate taken at random will he greater than 28.55 or less than
1V.75.
(h) Determine d/tr f or P = 0.01, 0.05, and 0.;30.
2. Suppose the odds in a 1way table are 87:1. Transform them to a 2 way basis.
3. In a wheat variety test, yields in bushels per acre were as follows:
Karired : 54.6, 53.7, 68.0, 55,2, 58.5, 62.1, 56.7
64,
.2, 57.5.
= 53.3
Cheyenne: 66.3, 60.9, 64.3, 67.6, 63.8, 62.2, 63.4
60.6, 67.2, 55.3, x = 64.3
Calculate: (a) The standard error for a single plot (s), and the standard error
of the mean (a g) for each variety; (b) The standard error of the difference
between the two varieties (07^); and (c) Determine whether or not the difference
between the varieties is statistically significant. Assume the population stand
ard deviations are different.
4. The yields of two varieties in bushels per acre are as follows for several repli
cations:
Variety A: 58. 40,40,42,39,35,32/26,42, and 44.
Variety B: 37,J>7,40,40,32, 30, and 31.
Compute a pooled estimate of the standard error (s) of the two varieties, compute
t, and determine whether or not the varieties differ significantly in yield by
reference to Table II in the appendix.
3 Two varieties of small grain, Big Four and Great northern, were grown each year
in adjacent plots from 1912 to 1020. The yields are given below.
Yields in Bushels per Acre
Year
Great Northern
Bis Four
1912
1913
191.4
1913
1916
1017
1918
1919
1920
71.0
739
48.9
78.9
43.5
4 ( . V
63.O
48.4
43.1
54.7
60.6
45.1
71.0
40.9
45.4
534
41.2
44.8
Which varieties yield higher? Is this difference significant •
(a) means of two independent samples? (b) Paired Samples?
shown by:
69
6. The grain yields in grama per plot for spring wheat irrigated at the tillering
and jointing stages were as follows for 1921 to 1923 (incl.):
Year
Plot
Tillering
Jointing
1921
A
B
C
D
155
232
2^3
257
281
202
271
265
1922
A
B
C
D
1+59
332
3to
312
366
)+o8
396
366
1923
A
B
C
D
513
5oi
563
3U6
602
635
593
539
3 Year Average
360
too
Determine whether or not irrigation at tillering results in a significantly higher
yield than irrigation at the jointing stage. Consider the values paired.
CHAPTER VII
THE BINOMIAL DISTRIBUTION AMD ITS APPLICATIONS
I . The Binomial Distribution
Suppose that "p" is the probability that an event will occur in one trial, and "a"
the probability of failure of that event to occur. Then., it can be shown by means of
the two theorems on probability that the successive terras of the binomial expansion
will give the respective probabilities that, in "n" trials, this event will occur
exactly N, N  1. N  2. ..... or times. The binomial expansion is as follows:
(p 4 q )N  p N + jj . pNlq ... H(Nl)
1.2
<l
,q N
h N(Nl)(N2 )p N ^ + .,..,.
1.2.3
where evidently p + q. = ]
Then, the probability of exactly X occurrences in N trials is;
X I N  X!
where NI =1.2.3 N.
This expansion is called the Bernoulli series or distribution. When p  q_, the
binomial distribution is symmetrical. This distribution is similar to the normal •
distribution for large values of N but it is unsuited for continuous variables be
cause the distribution itself is discontinuous.
(a) Eei ation to Probability
Suppose a die is thrown 20 times. In this case, the 21 terms of the expan
sion (l/6 + 5/6) will give the various probabilities of a particular face, say six,
appearing 20, 19, 18, 17 ....... or times.
Now suppose that the problem is more complicated. Let h dice be thrown 20
times and the sixes counted that appear on each throw. In any one throw the prob
abilities of getting i+,3,2,1 or sixes are given by the terms of (1/6 + ~)jo) L ~' r , To
secure the most probable results of the experiment, multiply each of these probabili
ties tr 20. The probability for sixes,
P/M = 1/1296 times 20 = 0.016
p/,\ = 20/1296 times 20  O.308
P/ 2 ) = 150/1296 times 20 => 2.J10
? (1} = 5OO/1.296 times 20 = 7.7IO
?f \ = 625/1296 times 20  9. 61+0
Then, the most probable outcome of the experiment is: No sixes, 10 times; 1 six,
8 times, 2 sixes, 2 times; J sixes, times; and h sixes, times.
( b ) Constants of th Bino mial Distribution
The formulas for the more important constants of the binomial distribution
are as follows :
70
71
Mean number of occurrences, x = Np(i)
Variance, o^ = Npq  (2)
Standard error, a = JWpq (3)
Probable error, P.E. = * 0.67^5 ^/Npa  (k)
Mean proportion of occurrence, p = Nipn + N 2 p 2  (5)
N x + N 2
Variance of proportion of occurrences, a*  pq   (6)
N~
Standard error of proportion of occurrences, o~ = /pq    (7)
II. Applications of the Binomial Distribution
The Binomial distribution may have a variety of uses in comparisons of observed data
with an a priori hypothesis or the comparisons of two samples.
(a) Comparison of Observatio ns against an a Priori Hypothesis .
Suppose in a sample of N trials of an experiment the number of occurrences of
a given phenomenon is x. Let it be desired to test this result in accordance vith an
accepted standard outcome of such experiments, the expected proportion of occurrences
being p. The expected number of occurrences (X = Up), and the discrepancy will be the
numerical value of x  Np = d. Then the probability that corresponds to t = d/V =
d/VNpq. may be found vith the aid of the t table when N is small, or with the table
of normal curve areas when N is large. It would be equivalent to test the proportion
(x/n) against the expected proportion (p) by the formation of t = d/cr = (x/n)  p
/Si
V N
A very common application is the comparison of observed data for monohybrid Mendelian
ratios with the theoretical. (See III below).
(b) Comparisons of Samp l es from Different Pop ulations
It may be desirable to compare the proportion of occurrences in two samples
from admittedly different populations. Then the samples provide the following infor
mation:
Sample I Sample II
N l Number of cases or trials N 2
x;l Number of occurrences of a x 2
given phenomenon
pj_ = X]_ Proportion of occurrences ]g>= x 2
N x Ng"
o~2 = p. q. Variance of the proportion cr^ = P2I2
~N7~ N 2
The differences in proportions = d = pi  pp.
The standard error of the difference = 0& = J^^ +
°o 2
Pl^l + P2<l2
n x iT~
72
Thus, t = d = pi 112
?]_q.]_  p 2 ^2
N x "No"
( c ) C omparison of Samples from Same Po pu lations
The difference "between this problem and that in (To) above consists in the
hypothesis that but a single population is being considered. As a result, the data
afforded by both samples are combined to give estimates of p and a for the population.
Thus, the estimated proportion of occurrences will be: p = NiP^ + N2P2, an( ' ^ ie
N, + N.
HO
estimated standard error of the population will be s e= mq_ , where q_ = 1p.
7 IT i  ; " H 2
Then t = pi  p 2 may be interpreted as in previous cases .
s
III . Standard Er rors of Men del i an Ratio s
In the analysis of genetic data, it is necessary to test the significance of the ob
served with the calculated counts obtained when certain theoretical conditions are
postulated. With monohybrid ratios, the general practice is to use the binomial dis
tribution, which is sometimes referred to as the probable error of a proposition.
The ratios which may be calculated in this work by the binomial distribution are:
1:1, 3:1, 9:7, 13:3, 15:1, 63:1, and 27:37.
(a) F ormula for M endel ian Eatios
The standard error of a Mendel ian ratio is:
o = Vp (1P)N or 751  ' (8)
where K = the number of individuals, p = the proportion of one group as a decimal
fraction, and 1  p = the proportion of the other group as a decimal fraction,
(l  p  q.) . Some writers use the formula, S.E. =yp.q.N, where "p" and "q_" represent
the proportions in decimal fractions.
( " b ) U se of Method
The binomial method can be used in genetic data only when two phenotypic
classes are grouped, other methods being used for three or more classes. In the Fv
generation of a barley cross, 200 green and 72 white seedlings were counted. It is
desired to test these data for a 3:1 ratio.
Gree n White Total (i Q
Observed numbers 200 72 272
Calculated 3:1 ra tio 2 ok 63 07?
Deviation '+
To obtain the calculated number for a 3:1 ratio, divide the total number observed by
the combined possible number of classes which is h in this case, e.g., 272 /k = 68.
This gives the calculated value for the white (or l) class. For the green (or 3)
class, multiply 63 by 3. This gives 204
a" = VpTip) n = Vo.75 x 0.25 x'272 * 7,]J+6o
Next., the deviation divided by the standard error is computed:
d/o = '1/7.1^60 = O.56
183
161
Jkk
258
66
3kh
75 3.0356
9.33
1935
150.5
3kk
10.5 9.2068
1.14
T3
The observed ratio fits the calculated 3>1 ratio very well, indicating that a simple
Mendel ian factor pair is responsible for the production of green and white seedlings
It is to "be noted that d/o is less than 2, which indicates that the fluctuation of
the observed ratio from the calculated may be considered as due to chance. In any
event, there is no reason to reject the theoretical ratio hypothesis.
Another example may be given for green and white barley seedlings.
Green White Total & 2 &/<7,
Observed
Calculated 3>1 ratio
Calculated 9*7 ratio
It is apparent that the data do not fit a 3:1 ratio as shown by the high value of
d/cr. However, they fit a 9* 7 ratio very well, indicating that there are two factor
pairs involved in the production of green vs. white seedlings in this cross.
(c) Short Cut Tables for Computations
Tables published by Cornell University give the Probable Errors for Values
of N from 11 to 1000. Another set of tables occurs in "Mendel ian Inheritance in
Wheat and Barley Crosses," by Kezer and Boyack (1918). The probable error values
obtained from such tables can be converted to standard errors by the division of the
probable error value by the factor, 0.67^5.
IV. The Poisson Distribution as A Special Case
As a rather special case of the binomial distribution, there is an approximation of
what is known as a Poisson distribution. This occurs when p, the probability of the
occurrence of an event, is very small and N, the number of trials, is very large so
that Np becomes appreciable. In a Poisson distribution the probability, Pv, of
exactly x occurrences in IT (IT = very largo) trials, is given by:
P x m e (Np)
x
i
Where e is a constant (2.718) and p is the probability of occurrence in a single
trial, and x\ = 1.2.3 x.
Although there are tables published of those probabilities, their use is unnecessary
in the more coirmion types of application.
(a) Constants of the Poisson Distribution
For the Poisson distribution, the moan and variance are equal.
Mean = x = Np
Variance = cr^ = Np, so that a = /Np
(b) Use of Poisson Distribution
The Poisson distribution gives a basis for the solution of many problems
that involve the maintenance of certain standards. Suppose that registered seed
regulations state that red clover seed must not contain over a given percentage of
noxious weed seeds in order to gain certification. Suppose that from a lot of seed,
a sample is taken of such size that a count of 10 noxious weed seeds corresponds to
the allowable percentage. In this case, the mean x = Np = 10. The standard error,
a = /Np = VlO = 31 However, 18 weeds may have occurred in the sample analyzed.
The whole lot is rejected for registration because the deviation from the mean,
18  10 = 8, exceeds twice the standard error, i.e., 2 6  6,2. Suppose that lb
noxious weed seeds are counted in a sample from another lot. Now a. decision "becomes
questionable. Suppose that a second sample is taken and Ik weed seeds counted.. How
consider the two samples as one. The mean., x  ftp = 20, and the standard error,
cr =/Wp =V20 = k. c ). Thus, ik + 16 = 30 which differs from the moan "by 10. How;
ever, this lot would be rejected "because the deviation from the mean, 10, exceeds
2 cr= 2(1*.. 5) = 9.0.
Reference!
1. .Anonymous. Tables of Probable Error of Mendelian Ratios. Department of Plant
Breeding, Cornell University ' (mimeographed) .
2. Fisher, R. A. Statistical Methods for Research Workers "(5th edition), pp. 5572,
19;A.
3. Kezer, Alvin, and Boyack, B. Mendelian Inheritance in Wheat and Barley Crosses.
Colorado Exp. Sta. Bui. 2k$ . 1918,  ./; •
h. Miles, S. R. A Very Rapid and Easy Method of Testing the Reliability of an aver
age and a Discussion of the Normal and Binomial .Methods.
5. Robertson, D. W. The Effect of a Lethal in the Heterozygous Condition on Barley
Development. Colorado Exp. Sta. Tech. Bui. 1. 1932.
6. Sinnott, E. W., and Dunn, L.. C. Principles of Genetics, , McGrawHill, pp. 371 375 ■
1932.
7. Tippett, L. H. C. The Methods of Statistics. Williams and Nor gate, pp. 3033
1951.
Questions for Discussion
1. Give the binomial expansion of (p ■< q) .
2. What type of distribution is the binomial distribution? What are its limitations?
3 What is the genetic application of the binomial distribution? Its limitations?
k. How does the Pois^on distribution differ from the binomial distribution?
3. Under what conditions might the Poisson distribution urove useful?"
Problems .
1. Colsess, a whiteglumed barley was crossed with Nigrinudum, a blackglumed barley.
The segregation in the Fo was 785 blackglumed plants and 215 whiteglumed plants.
What ratio best fits these data? Calculate a/a.
2. The Eg segregation of a Colsess (hooded) by Minnesota 908 (awnod) cross gave 229
hooded plants and 89 awnod plants. Determine the ratio that bsst fits these data,
and test its fit.
3. In a cross between Colsess II and Colsess III, 183 green seedlings and 161 white
seedlings were observed in the Eg. Determine the ratio that best fits these data
and test the fit .
CHAPTER VIII
THE X 2 TESTS FOR GOODNESS OF FIT AM> FOR IMPENDENCE
I. ThcX 2 Test
So far, statistics like the sample mean (£) and the standard deviation (s>) have been
used to express differences between distributions, either an observed against a
hypothetical distribution, or one observed distribution against another. However, in
such cases the general form of the distribution (normal, binomial, Poisson) has been
assumed and comparisons have been limited to values of parameters of the distribution.
The use of moments such as these might be adequate for an accurate comparison of dis
tributions were a sufficient number of higher moments employed. However, this method
has the principal disadvantage of being tedious as well as involving questions as to
the validity of the sampling errors of higher moments.
Many times it is desired to compare or te3t observed data with those expected on the
basis of some hypothesis. This has been referred to as a test for "goodness of fit."
Again, individuals may be measured or classified categorically with respect to two
separate characters or conditions. It may be desired to test these characters for
association. Both of these general problems can be attacked by use of a statistic
known as X 2 (Chisquared) calculated from the data afforded by the sample.
II. The X2 Distribution
The X 2 test, to measure "goodness of fit" of observed results to those expected, was
advanced by Karl Pearson in 1900.
(a) Formula for X 2
The theoretical distribution must be adjusted to give the same total frequen
cy as the observed. Then, when is the number observed in any one group or category
of the experimental distribution, and C the theoretically calculated number for the
same group, based on the hypothesis that the data follow some certain distribution,
the formula f or X 2 is as follows:
2
X = S
J (1)
(0  c) 2 1
c
where "S" is the summation extended over all the groups or classes. It is obvious
that the more closely the observed number agrees with the calculated the smaller X 2
will be. Further, all differences in frequency (0C) are squared, whether positive
or negative. Thus, X 2 is always a positive quantity, its size being clearly dependent
on the number of groups into which the distribution is separated and degree of agree
ment between the several values of "0" and the corresponding values of "C". There
fore, in the ordinary application of X 2 , the number of degrees of freedom will be the
number of groups diminished by the number of restrictions imposed on the theoretical
distribution that supplies the values of C . When the only restriction imposed is
that the total frequencies of the observed and theoretical distributions shall be
equal, the degrees of freedom are one less than the number of groups. In other words,
where the frequencies are determined for all groups but one, the frequency of that
one is automatically determined by subtraction from the total.
(b) Sampling Distribution
The sampling distribution of X 2 has been worked out so that it is possible to
find the probability (P) of obtaining from a hypothetical population with a given
distribution, a sample that shows a distributional variation from that of the popula
tion which would result in a X 2 value as large or larger than that exhibited by the
sample in hand. For every value of X 2 ; in conjunction with any optional number of
75
76
degrees of freedom P = 1.00 f or X 2 = and, as "X5 increases, P diminishes. Sirce the
mathematical relationships between X 2 and P are complex, it is necessary to have
tables that relate P, X 2 , and the number of degrees of freedom, for practical use.
( c ) Grouping Da ta
It is unwise to group too finely or to apply this test where the data are so
insufficient that, for certain of the groups, the expected frequency is small. This
condition very easily might cause that part of a 2 contributed by such groups to un
duly affect the total X 2 , This is obvious from the mathematical form, (0  C) r /C ;
where C is small. Fisher (193*0 recommends that each group should contain at least
five individuals for the test to apply. Sometimes the tail groups with very low
frequencies should be combined.
111 • Prob ability Tables for X 2
o _
As has been stated the probabilities for X c  values are obtained from tables . m
'order to use them, it is first necessary to know :, n", the number of degrees of free
dom in which the observed series may differ from the hypothetical. It Is equal to
the number of classes, the frequencies in which may be filled arbitrarily. When only
the totals, have been made equal, n = n'  1, where n.' is the total number of classes
or groups. In contingency tables, where tests for independence are being made, the
number of degrees of freedom is the product of rows and columns minus one in each
case (r  I) (c  1) because the hypothetical and observed classifications are forced
to conform both for row and column totals. To quote Tippett (1931): ''Suppose, in an
extreme case, there are n' groups and we fitted a curve Involving n' constants which
were calculated from the data.; then the two distributions would agree exactly and X^
would be zero because sampling errors would have had no play," The importance of
degrees of freedom in looking up the probabilities that correspond toX2 has been
emphasized by Fisher (1922, 1923, 193*0 •
(a) Elderton "Table of Goo dness of Fit"
A table was prepared by Elderfon with the values of "?" (probability) that a
deviation as great as or greater than the observed. may be expected on the basis of
random sampling. These values correspond to each integral valtIe>or ^ from Fto 30.
This table is available in "Tables for Statisticians and Biometricians" by Karl
Pearson (191*+) • The user must be careful with bhis table because n' is equal to the
number of degrees of freedom (n) plus one. The probability of Intermediate X.' values
can be obtained approximately by interpolation.*
(b) Fisher "Table of X 2 "
More recently, Fisher (193*0 ^ aG published a table of X 2 which uses degrees
of freedom (n) directly. It gives values of X 2 that correspond to special valv.es cf
"?". Fisher (193*0) states: "In preparing this table we have borne in mind that, in
practice, we do not want to know the exact value of 'P' for any observed X2, but in
the first place, whether or not the observed value is open ; :o suspicion. If r P' is
between 0.1 and 0.9 there is certainly no reason to suspect the hypothesis 'tested.
*Note: For example, the probability forX 2 = J+.12 determined from h classes can be
Interpolated as follows:
WhonX 2 = k, P  0.26lhbk
k. 12
X 2 = % P ^ 0. 171797
Difference 0.12 O.OG9667
Product 0.12 x O.O89667 * O.OlO'JoO
"?" value 0. 2611*64  O.OIO760 = 0.25070^
77
If it is below 0.02 it is strongly indicated that the hypothesis fails to account for
the. whole of the facts. Ve shall not often go astray if we draw a conventional line
at 0.05 and consider that higher values of X 2 indicate a real discrepancy." The
table given by Fisher has values of "n" up to 30. Beyond this point it will he found
sufficient to assume that y^X 2  y2nl is distributed normally with unit standard
deviation about zero. For example:
X 2 = 35.62, n = 32, V2X 2 = 8.^,72nl = 7.914, Difference = 0.50.
Thus, where /2X 2 /inl is materially greater than 2, the value of X 2 is not in
accordance with expectation.
(c) Normal Probability Integral Table
In the special case for one degree of freedom (n  l), the probability can
be obtained from the table of the normal probability integral because X is normally
distributed for one degree of freedom. (See Table II, "Tables for Statisticians and
Biomecricians") For example, suppose it is desired to find the probability that
corresponds to X 2  3. 200
X=/X"2" ^/J^OO n 1.7639
In the table opposite t = 1.7889* the value of the probability that corresponds to it
is found to be O.9632. The value of the probability for the one tail will then be
1.0000  O.9632 = O.O368. On the basis of a 2 tailed table it would be O.O368 x 2 =
0.0736.
A  Goodness of Fit
IV. Uses of X 2 for Goodness of Fit
The X 2 test for "goodness of fit" can be applied to data grouped into classes where
it is desired to compare them with a theoretical or hypothetical ratio. The great
advantage of this test for goodness of fit is that no limitations or conditions are
imposed upon the form of the distribution under investigation. Historically, the X 2
test was first ured to test the goodness of fit of an observed frequency distribution
to a normal distribution of the same total frequency, the same mean, and the same
.standard deviation. It is still used effectively for this purpose when the number in
the sample is large. One sacrifices a fit in the tails of the distribution by use of
the X 2 test, but often the investigator is only interested in the central range which
the data cover. The X? test is particularly useful in genetics to test Fg and later
segregations where two or more phenotypic classes are involved. J. Arthur Harris
(1912) first called attention to the value of the X 2 test for genetic data.
V. Computation of X 2 for Goodness of Fit
In Mendelian ratios from F2 progenies and later generations, the common practice is
to summate the numbers in each phenotypic class and to formulate a hypothesis on the
basis of the ratio obtained in order to establish the number of genetic factors in
volved. TheX 2 test is used to determine whether the deviations of the observed num
bers from the calculated numbers are not due to chance.
(a) General Method of Computation •
In a cross that involves two independently inherited Mendelian factor pairs,
a 90:3:1 ratio is expected in the F2 generation. A segregation in the F2 generation
of a barley cross that involved long vs. shorthaired rachilla (Ss) and covered ve .
naked seeds (Nn), gave results as follows: (Data from Bob ert son)
78
Long Haired Baohilla Short Haired. Rachil la Total
Covered Seeds Naked Seed s Covered Seed. a , Naked Seeds
2061 6I+5 673 256 ' 3637
(SN) (Sn) (bN) . (sn)
The calculated ratio for a 9:3:3:1 is calculated so that the total of the theoretical
values equal the total in the sample,, i.e., 3637 • The value 3637 is divided hy 16
(Q +3 +3+1) to give the expected number in the class with shorthaired rachillas
with naked seeds, i.e., 3637/16 = 227. 3125'. The values on the basis of expectancy
for the 3classes can be computed "by multiplying 227.3125 by 3 = 681.9375; etc. The
results can be put down as follows:
Observed Calculated „
Classes Ratio No. (0) No. (C)  C (0 . C) 2 (0  C) d /C
SN 9 206l 20145.81 I5.I9 23O.736I 0.1128
Sn 3 645 ' 681. 9k 36.9k x3.64.5636 2.0010
sN 3 675 68l,9'4 6.9*4 kQ. 1636 0.0706
_sn_ 1 236 J227 ._1 _ _ 28 . 69 823. 116 1 __. 6211
Totals 3637 3637.OO X 2 = 5.3055
n = 3 P« 0,lcJ33
Hence } the deviations from the calculated ratio cannot be regarded as significant.
(b) Method fo r Two Classes
The X.2 value may be calculated directly where "A" is the number in one class,
"a" the number in the other, and "N" is the total number in the sample (A + a) .
These formulae are given by Immer (1936) and represent a transformation from the
standard method for the computation of X.2 for goodness of fit.
pati o A : a X Va lue
(A  a) 2
1:1 N  ..__..._ (2)
3 : 1 (A _ _g_2 _ _ „ _ _ „ „ ........  (3)
3N
9 : 7 (7A  9a) 2 . _... (M
631^
m : n (nA  m a) 2 •___„___._„__..________„  ( 5 )
mnil
The computation may be illustrated with data which appear to fit a 5 : 1 ratio.
A = 2903, a = 936, and N = 3839.
"* 2 = (A  5e ) 2 = (2905  280o ) 2 = 0.7840. ? = Value close to 1.
3N 3 x 3339
( c ) The~X2 Test Applied t o S ev era l. Qe ne tic F ainil ies
In genetic data, Kirk and Immer (1928) show that the total class frequencies
obtained by summation are composite results which may easily mask a serious lack of
consistency in numerical ratios of the separate families with respect to agreement
with expectation. To summate the numbers in each class of all progenies is to rely
on mean values and thereby disregard deviations from the ratio expected to occur in
each family. This applies particularly where the numbers are small. The smaller the
number in each progeny, the greater the opportunity to err when the summations are
taken as an indication of the genetic constitution. In such cases, a goodnoss'of
fit test like Xj2 ± B required which involves in its calculation deviations from ex
pectancy for each class of each progeny. It should be mentioned thatXA values can
79
not be averaged. However, they are additive provided the number of degrees of free
dom are properly taken into account.
VI. Fit of Observed Data to the Normal Curve
The \2 criterion is useful to determine whether or not observed data give an accept
able fit to the normal curve or any other assumed form of distribution. It is useful
where the sample is large and where the requirements f or X2 are fulfilled. First,
the range of measures is divided into an arbitrary number of classes so as to meet
the number of measures in the separate classes which a valid use of the X 2 criterion
demands. Data on number of culms counted on 1+11 wheat plants at the Colorado Experi
ment Station are used to illustrate the computation. The data are as follows:
X (Class center) 1
f (Frequency) 2
3
5
7
9
11
13
15
17
2^
52
85
Ufc
69
36
18
6
19 21 23 25 27
1 2 1 1 = 1+11
2 = 8.9172 s' = 3.U715. s' (corrected for grouping) = 3.1+231.
(1) The data are regrouped in order to have a larger number of cases in the tail
classes.
Classes
less than 1+
7
> 9
11
13
15
more than 16
Class Range
1+.0 to 6.0
6.0 to 8.0
8.0 toiao
10.0 to 12.0
12.0 toll+.O
ll+.O to 16.0
2b
52
85
111+
69
36
18
11
The class range is reduced to
cr units, viz., 2/3.1+231 = 0. 581+3
The correction to the mean above
8.0 = 0.9173/3.^231 = 0.2680
cr units.
II =
1+11
(2) The next step is to calculate the end points of units for the class intervals in
ct unit s .
Unit
Calculated
Class rang
e
1+.0
Area Range
Frequency
Frequency
Less than

OO T,0
 1.1+3
0.08
32.9
1+.0 to.
6.0

1.1+3 to
 0.85
0.12
1+9.3
6.0 to
8.0

O.85 to
 0.27
0.19
78.1
>8.0 to
10.0

0.27 to
+ 0.32
0.21+
98.6
10.0 to
12.0
+
O.32 to
+ O.90
0.19
78.1
12.0 to
ll+.O
+
0.90 to
+ 1.1+8
0.11
1+5.2
ll+.O to
16.0
+
1.1+8 to
+ 2.06
0.05
20.6
more than
16.0
+
2.06 to
+ 00
0.02
8.2
Total
1+11.0
The cr  value for the class that contains the mean is: 0.581+3  0.2680 = +O.3163 cr.
This value is the ordinate for 10.0 while 0.27 is the cr ordinate for 8.0, these
values being within the range, 8.0 to 10.0. The other area ranges are calculated by
the addition of O.58 to determine the next higher or lower range. For example, it is
0.32 + O.58 = + 0.90 for the range 12.0.
(3) It is now necessary to compute the unit per cent frequency for each class by
reference to a table of the probability integral, (Table I, appendix) . For exam
ple, the unit frequency for the range, 0.27 + O.32 is computed as follows:
For t = 0.27,
P
= 0.61
 0.50
= 0.11
t = +0.32,
P
= O.63
 0.50
= 0.13
8o
The frequency per cent for the distance ^ 0.27 to +0.32, is equal to 0.11  1  0.13=
0.24. This means that 24 per cent of the frequencies would "be within this range
or the basis of the norma], curve. The other values can he calculated in a simi
lar manner , except that the two values are subtracted. The last class, from 2.06
to include the remainder of the curve, is computed as follows:
For t = 2,00, P = 0.98 1.000  O.98 a 0.02
(•4) The next step is to multiply the per cent frequencies "by the number in the sample
(N) to obtain the calculated frequencies., e.g., (0.08) (411) = J2..9, etc.
(5) The observed and calculated frequencies are now compared by use of the X> cri
terion.
Observed Calculated
Class range Frequency Frequency 0C (0C) 2 (0C) 2 /C
less than 4.0 2b 32.9 6.9 47. 6l 1.4471
4.0 to 6.0 32 49.3 2.7 7.29 0.1479
6.0 to 8.0 85 '73.1 6.9 47.61 O.6096
8.0 to 10.0 114 98.6 13.4 237.1.6 2.4033
10.0 to 12.0 00 78.I 9.1 82.81 . I.O603
12.0 to 14.0 36 43.2 9.2 84.64 ■ I.8726
14.0 to io.o 18 20.6 2.6 6.76 0.3282
mer e than 16. 11 _ 3 « 2 _ _2'2_ 7e4 . O.o^ol
Totals 411 411.0 • X 2 = 3.8271
P  O.II72
There are 8 classes, but only 3 degrees of freedom available because 3 constants have
been used in fitting the da.ta to the normal curve. lb is obvious in this case that
the probability (P) is greater than 0.05 . Thus, the underlying distribution of the
data may have been normal. This method, applies to fitting observed data to any
hypothetical distribut ion .
'VTI. Part ition of X 2 into it a_ Co mpo nent s
When a discrepancy in a theoretical genetic ratio on the basis of independent inheri
tance occurs, it may be produced either by linkage or a departure from the 3 : 
ratios. Fisher (193*0 bias suggested a method whereby X 2 can be partitioned into its
components to determine the source of the discrepancy. In a barley cross, the Y,p
data were as follows for nontipped and tipped lateral spikelets (Tt),.and for hoods
and awns (Kk) :
TK Tk tX tic Total
" (a) ' ~ (b) ' "(c)" (d)
Observed No. I496 315 550 216 2777
Calculated No. 1 562 .'06 ■ 520. 69 520. 60 173^6 2 77700
X.2 ^ 14.8855 P  very small
To determine whether or not the discrepancy is due to linkage, the "X.2 value is par
titioned into its components as follows:
x = nontipped vs. tipped => (a + b)  3( c + d)
 (1J4.96 + 515.)  5(350 4 216) = 287
y = hoods vs. awns = (a 4 c) ■ 3(1 + d)
= (1496 + 550)  3(513 + 21b) * l'i7
z = interaction or linkage = a 3b  3c + $'d
 1496  3(515)  5(530) v 9(216) * +243
81
Thex2 values can be computed for each component as follows:
1. nontipped vs. tipped:
X 2 =jc£ = (287) :
3n
3(2777)
2 . hoods vs . awns :
X 2 = y£ = (l^T) 2 .
3^" 3(2777)
3. interaction (or linkage):
.X? =_z?_= (243) 2
9n 9(2777)
= 9.8870
= 2.5938
= 2.1+017
The data can he brought together in a summary form as below:
Factor Pairs d.f .
X 2
Nontipped vs. tipped (Tt)
Hooded vs. awned (Kk)
Interaction
9.8870
2.5938
2.1+017
0.0016
0.107^
0.1212
Totals
11+ .8825
very small
Thus, the 3 • 1 ratio for nontipped vs. tipped is found to account for a Large part
of the high X2 value. There is no indication of linkage.
B  Test for Independence
VIII . Independence and Association
When observations have been classified in two ways, it may be desirable to determine
whether or not the two variables are associated. The%2 test for independence has
been used for this purpose. Two variables are said to be associated when the numbers
in the cells of the contingency table are not randomly distributed. Contingency
tables may be manifold, there being (r  1) (c1) degrees of freedom where there are
"r" rows and "c" columns. In tests for independence^ .the subtotals of the classes
into which the variates are distributed are used to determine the theoretical fre
quencies with the result that the subtotals, must be considered as constants in the
determination of degrees of freedom. For example, the degrees of freedom in a 2 by
2 contingency table are one . The value of X2 is referred to a X 2 table to determine
the value of "P" that corresponds to it for the number of degrees of freedom in the
contingency table. A "P" value greater than 0.05 indicates lack of proof of associa
tion between two variables, i.e., they may be independent. The x2 criterion has
proved useful as a test for the independence of two genetic factor pairs.
"DC . Calculation of Independence or Associat ion '.
The test for independence can be made when the data are compiled either in simple
l+fold (2 by 2) or manifold contingency tables.
(a) The Manifold (m by n) Contingency Table
The computation can be illustrated by some Fg data (Hayes) in an oat cross,
Bond x D.C, where it was desired to learn whether or not there was any association
between the reaction to stem rust and to crown rust. The data are:
82
Stem Rust Reaction
Resis tant Susceptible Totals Ratio
Crown
Rust
Reaction
Resistant
Susceptible
Intermediate
50 (57. 2 W
119(112.1126)
2 5(24. 658O)
22(14.7550)
22(2,8.8950)
6(6.5500)
72
141
•51
0.2951
0.5779
0.1270
Totals
194
50
244
1.0000
In case that the amount of stem rust infection has no influence on the amount of crown
rust infection,, the 244 observations would bo expected to be distributed at random in
the 6 cells of the contingency table, with the restriction that they must add up to
give the totals in the table (See Tippett, 1951., p. 69). The probability that an
observation will fall in row No. 1 is "(2./2kh, and that it will fall in column No. 1
is 194/2.44 . Then, the probability that an observation will fall in the first cell Is
(72/244) (194/244). The expected number of individuals in that square on the basis
of independence is the probability multiplied by the total number, i.e., (72/244)
(iqh/2kh) (244) = 57,2^94.
The various steps in the computation are as follows:
(1) The ratio of rows, for row No. 1, is 72/244 = 0,2951.
(2)' The theoretical frequencies can be obtained by the multiplication
of each of the ratios for rows by each of the subtotals for columns,
e.g., 0.2951 times 194 = 57.2494 for cell No. 1. The other values
are computed in a similar marine]
In this case, it is necessary to
compute the value for only one other cell, i.e., 0.5779 times 194 =
112.1126. The other values can be obtained by subtraction from the
marginal totals .
(3) The observed and theoretical values are then compared by use of the
Xr criterion.
01
served
Calculated
Nc
.
No.
0C
(0C) 2
(oc) 2 /c
50
57.2494
7.2494
52.5538
0.9180
119
112.1126
6 . 8874
47 .if 363
0.4231
25
2k . 6580
0.5620
0.1310
0.0053
22
1.4.7550
7.2^50
52.4900
3.3574
22
28.8950
6.8950
47.54IO
1.6453
6
6.5500
0.3500
0.1225
0,0193
244 244.oooo X2  6.5684
n = (n  1) (m  1) = 2 P = O.O387
Thus, the indications are that there. is an association between the reactions to stem
rust and to crown rust.
(b) The 2 by 2 or *4Fol d Table
The 4fold table Is often used to test tho independence of two genetic factor
pairs. The independence of the two 3 : 1 ratios can be tested as follows:
K.
k
Total
V
v
a = 142
c ~ 49
b  4j
d = 15
a + b = 185
c •* d = 64
Totals
a + c • 191 b + d = 58
N
_ oiir
y
The value of x2 can be determined by the method outline in (a) above, or it can be
•■coiaputed by a shortcut formula given by Fisher (193' 4 ) •
■2 _
U
N (ad  be) 2
c) (b + d) (a
+ b) (c + d)
83
(6)
■ 249 Ha^)(g),,:.,(^)^9)3
U9lJ(58)(l85)(5IsT
(2^9) (529)
(191)(58)(185)(6U)
151,721
131,163,520
0.0010
whenX 2 = 0.0010, P = value close to 1.
(c) Inadequacy of X 2 : Correction for Continuity
When the several categories are represented by relatively small frequencies,
the value of X 2 often gives inaccurate results because the corresponding probability
of occurrence is too small. This is particularly the case in a 2 by 2 classifica
tion. Yates (193^) &as developed a correction that should be applied in such cases.
This correction simply amounts to the reduction of each numerical value of each (0C)
determination by l/2. Thus, in the example above, the correction applied to "X 2 is as
follows:
"X (corrected) =
N(ad  be  N/2) 2
(a + c)(b + d)(a + b)(c 4 d)
2k 9 C(3*2)(15)  (W(k9)  2U9/gl g
(7)
(19D(58)(l85)(6l + )
(2k 9 ) (IO6.5) 2
131,163,520
0.0215
 2,82^,220.25
131,163,520
P  value close to 1 .
In this case even tho the frequencies may be fairly large, it is quite proper to
introduce the correction. However, the larger the number of categories, the less •
important is the correction.
X. The Null Hypothesis andX 2
It is important to understand something about the philosophical and logical bases for
the making of inferences from the X2 as veil as from other criteria for significance.
The basic premise involved in every test for significance is a negative premise and
has been termed the null hypothesis by Fisher (1937) . It is simply a tacit assump
tion of agreement, such as agreement between standard deviations of distributions, and
agreement between distributions as a whole. In association and correlation studies
the null hypothesis is construed to mean independence or lack of association between
characters or conditions under investigation. This tacit negative premise can never
be proved. For example, it is impossible to prove statistically that two samples came
from the same population, or that the population which afforded the samples under
comparison possess the same means or other statistics. It is impossible to prove
statistically that two characters or conditions are independent or devoid of associa
tion. To draw such conclusions would simply be to reiterate what was originally only
assumed to be true.
Therefore, definite conclusions can be drawn only when the criteria for significance
have been met, such conclusions being positive in nature . The investigator is able
to prove differences to exist, association to be present, etc. In short, he is abl
to prove the falsity of the null hypothesis but never its truth.
34
Reg erences  : ' >
1. Fisher , B. A. On the Interpretation of X 2 from Contingency Tables, and the
Calculation of ?. Jour. Roy. Stat. Soc, Vol. LXXXV , Part I, 1922.
2. Statistical Tests of Agreement between Observation and Hypothesis.
Economic a, 3:139147. 1923.
3. Statistical Methods for Research Workers (5th Edition), pp. 80111,
and 274275 193^.
4 The Design of Experiments, pp. 1.820. 1937
5. Goulden, C. H. Methods 'of Statistical Analysis, pp. 88113 • 1939.
6. Harris, J. A. A Simple Test of Goodness of Fit of Mendellan Ratios, Amer, Nat.,
46:74l. 1912.
7. Xminer, F. R. Applied Statistics Manual (mimeograi:>hed) .. 1936.
8. Kirk, L. E., and Immer, F. R. Application of Goodness of Fit Tests to Mendelian
CI as 3 Frequencies. Sci. Agr., 8: 7'! •'>' ■■">'■) . L923Y
9. Pearson, Karl, Tables for .Statisticians and Biometri clans, pp. 23 and 2.628.
191^. . •■: ■•■:■
10. Tippett, L. H. C. The Methods of Statistics, pp. 6388. 1931
11. Yates, F. Jour. Roy. Stat. Soc, Suppl. I, Tio. 2. 1934.
12. Youden, W. J. Statistical Analysis of Seed Germination thru the Use of the Chi
Square Test. Cent rib. Boyce Thompson Inst., 4:219232. 1932.
13. Yule, G. Udny. An Introduction to the Theory of Statistics (9th edition),
PP. 370378. I929.
14. Yule, G. Udny. Probability Values for One Degree of Freedom. Jour. Roy. Stat.
Soc, 85:93104. 1922.
Questions for Discussion
1. What are the uses of the X 2 criterion?
2. What conditions must be fulfilled in the use of the X s test?
3. What Is the range of X 2 values? "P" values?
4. Give a rule for the number of degrees of freedom in a "goodness of fit" test.
What is it for a contingency table?
5. What precautions are necessary in the grouping of data for a "goodness of fit"
test? Why?
6. How do the Elderton and Fisher tables for X. 2 differ? What precautions are.
necessary in the use of each?
7. Interpret "P" = 0.50 on the basis of goodness of fit.
8. In what special case can the normal probability integral table be used to compute
"P"? Why?
9. Who is responsible for the X.2 test? For what was it first used?
10 . Explain how to compute X2 for goodness of fit.
11. What precautions are necessary in the application of the X 2 test for goodness of
fit to genetic ratios? Why?
12. In the fitting of observed data to that expected on the basis of the normal curve,
how many constants are used? Which ones?
13. Under what conditions may it be desirable to partition X 2 into, its components?
14. What does "P" = 0.01 indicate when obtained from a contingency table?
13. Explain how the probability is calculated for a cell In a contingency table.
16. How does the X2. test for independence differ from that for goodness of fit?
17. What is meant by the null hypothesis?' . ,,
85
PROBLEMS
I. In a "barley cross, Robertson (1929) tested black vs. white glumes (Bb) and hoods
vs. awns (Kk) for a 9 ' 3 '• 3 ' 1 ratio in the Fg. His data were as follows:
Classes Observed No. Calculated No.
Black hooded (BK) 2611 2656.7
Black awned (Bk) 920 885.5
White hooded (bK) 860 8P5.5
White awned (bk) 332 poc *.
■■■■ 1 ■ ■ 1 1 , ,■ 1 1 I 1 1 . ■ ii 1 ' ■ * * r — . —
Totals ^723 1+723
Calculate X 2 and interpret it.* Do these data fit a 9 '• 3 ' 3 : 1 ratio for
independent inheritance?
II. Some data on hoods and awns (Kk) and covered vs. naked (Nn) in barley were test
ed for a 9 : 3 : 3 : 1 ratio. The observed and calculated results were as
follows: (Data from Robertson, 1929)
Calculated No.
~2bTS
682
682
227
Totals 3637 3637
Apply theX2 test and interpret it.
III. An Fg segregation of a barley cross, Colsess x Minnesota 8U7, gave these re
sults: (Data from Robertson, 1929)
Classes
Observed No.
Hooded covered
(KW)
1969
Hooded naked
(Kh)
631
Awned covered
(kN)
737
Awned naked
(kn)
250
Classes
Obi
served No.
Hooded green
(KF)
931
Hooded chlorina
(Kf)
326
Awned green
(kF)
326
Awned chlorina
(kf)
119
(a) What ratio fits these dotal (b) Apply X 2 test and interpret it. (c) Calcu
late the probability both from the table by Fisher and from the table of
Elderton.
IV. In the F2 of a certain barley cross there were 2U9 plants with high fertility of
the lateral spikelets and 67 with low fertility. Test these data for a 3:1
ratio by the X2 test for goodness of fit.
V. A second generation segretation in a barley dihybrid for high and low fertility
(Hh) and for black and white glume color (Bb) gave counts as follows:
HB Hb hB hb
15^7 568 I+78 184
*Note: Statement for P: "A worse result might be expected on the basis of random
sampling times in trials . "
86
When these data were tested for a calculated 9: 3 5' 1 ratio, X 2 was 3.5718
with P = O.0365. Partition X 2 into its components and determine whether the
discrepancy is due to the individual 7 j> : 1 ratios or to linkage.
VI. Some Fg oat plants were classified on the hasis of crown rust and stem rust
■resistance as follows:
Stem Rust Reaction
Resi sta nt Susceptible
Crown Resistant 66 hj> 109
Rust Susceptible 75 "" 2k 99
Reaction Intermediate 17 5 22
Totals 158 72; ' , 2.30
Use the X2 test f 02 independence to determine whether or not there is an asso
ciation "between the reaction to stem rust and crown rust .
CHAPTER IX
SIMPLE LINEAR CO.REELATIQ K
I. Nature of Correlation
So far, statistical analysis has dealt with a single set of observations to measure
a single character. It is now desirable to consider two such sets of observations
that measure two different characters. These observations are such that, to any
observation in one set, there is naturally paired a corresponding observation of the
other. One naturally inquires as to whether there exists any association or connec
tion between the measured characters. Such association exists when an abnormality^
in one character tends to be accompanied by an abnormality in the other. The charac
ters are said to be correlated when such is the case. For example, height and weight
in human beings are said to be correlated. In the aggregate, tall persons are heavier
than short persons.
To condense what nas been said into a precise definition, it may be stated that two
characters are correlated when, to a selected set of values of one, there correspond
sets cf values of the other whose means are functions of those selected values.
II. Description of Correlat ion
A graphical representation of the totality of paired observations can be obtained by
the treatment of each pair of measurements as the rectangular coordinates of a point.
Such a diagram of scattered points is called a scatter diagram. To illustrate, one
may consider 20 pairs of observations that relate length (in inches) to weight (in
ounces) of ears of corn:
Length (x)
Weight (y)
Length (x)
Weight (y)
2.5
2.5
3.0
k.Q
k.5
5.0
5.5
6.0
6.0
6.5
3.5
3.0
5.0
7.0
5.5
8.0
8.0
10.0
T.o
10.5
6.5
7.5
8.0
8.0
8.0
8.5
9.0
9.0
9.5
10.5
b.5
10.0
8.0
10.0
12.0
13.0
12.0
1^.0
13.0
lh.O
Mean length (x) = 6.5 inches. Mean weight (y)  9.0 ounces
From these pairs of measurements a scatter diagram can be made as follows:
1 Abnormality refers to  deviations from the mean,
87
QP
d
Ear length (:ln.)
i+. 56 7 89 1.0
11
Ear
weight
(oz.)
5
6
I
3
9
10
11
12
13
1.1+
• • •
• • *
I
1 , • *
J = 90
x ■ b . '
From the diagram, it is clear that the horizontal and vertical lines that represent
the moan length and weight of the ears in the sample separate the plane, 'in which the
points are plotted, into four regions or quadrants. It is also evident that most of
the points fall into two of these regions, i.e., those which describe the abnormali
ties in regard to the characters to be of the same typo above the average and below
the average. Thus, there appears to exist a direct or positive correlation between
the characters .
The totality of points that form the scatter very often possess the rough geometrical
form of an ellipse. The position of the ellipse indicates the type of association,
i.e., whether positive (direct) or negative (inverse). The shape of the ellinse
roughly estimates the degree of correlation. The characters are closely related when
the ellipse is narrow. A diagramatic representation of correlation is given in
figures A, B ; and 0.1
ISome statisticians use the first quadrant in correlation analysis while others use
the fourth.
IB!
89
X 4
1
^
V
4 J
1 " '
\ +
N.
X.
+\
y
Figure A
Low Correlation
Figure B
Positive Correlation
Figure C
Negative Correlation
The signs for the quadrants are depicted in figure A. It is noted that the values
of x above the mean (x) are positive, while those "below the mean are negative. The
same applies for the y values. The sign for the quadrant is the product of the
corresponding marginal signs .
There are two methods employed to describe correlation, i.e., the correlation surface
method and the regression method. For an account of the correlation surface method,
a text on mathematical statistics should be consulted.
A  The Correlation Coefficient
III. Measurement of Correlation
A precise mathematical measure of the degree of association between two characters is
desirable. In any case, it must be based on an assumption in regard to the mathemati
cal functional relationship that exists between the variables. The most important
measure is called the coefficient of correlation, symbolized as r. In the discussion
that follows it is assumed that the association is linear, i.e., that the variables
x and y are related by an equation, y = ax + b, where a and b are constants.
Suppose one considers each pair of measurements of the two characters as an argument,
either strong or weak, for one or the other of two opposite theories of association
between the two characters. These theories are that the two characters are related,
either positively or negatively. A linear relationship is said to exist between two
characters when the moans of the values of one character are plotted with the selected
values of the other character that correspond to them so mat the resulting points
are well fitted by a straight line. To measure the contribution of any given pair of
measurements (x, y) to one theory of association or the otner, one measuresthe amount
cf abnormality exhibited by the pair of measurements with respect to each character
in units of the respective "standard deviation of the samples provided by the 2 sets
of variatea, i.e., (x  x)/s'y. When these measures of the abnormalities of the
pairs of observations are multiplied, i.e.,
(x  x)(y  y) " ' . ■•
s i s i the result gives a numerical measure of the argument presented
y by (x, y) toward a theory of correlation. The product of both
•abnormalities will be positive when both are of the same type, either positive or
negative. Their product will be negative when the abnormalities are opposite in type.
A numerical measure of correlation between the characters under investigation is ob
tained when the procedure is repeated for every pair of measurements in the sample
and the arithmetic mean of the several products is found. The formula for the correla
tion coefficient (r) is as follows:
90
r 
' i s r*— * \ / y  y \ .... • t .s
N I s' x / \ » y /  " * ' ' :"W
It is obvious that "r" can "be plus or minus, thus depicting a positive or negative
correlation. It will he shown later that "r" is numerically equal to or less than
1.0. Thus, the association that exists "between two characters may he strong, as
evidenced hy a value of "r" numerically close to 1.0, or weak when "r" is close to 0.
The above statement must not he construed too literally' hut in the light of sampling
theory.
IV. Computation of "r" for Ung r ouped Data
The relation, r = _1 S •' x  x \ / y  y A may he transformed to many &iffe?reii v
arbitrary forms for computation. Formulas which are useful for .email samples are as
follows :
r X = IJja O  N x y
/(Sx* " Nx 2) ( Sy 2 I N =2) '
r = S(xy)/N  x y
r  NS(xy) : (Sx)(&y ) ^
/[>3(x 2 )  (Sx)2J [NS(y2) (Sy) 2 ]
(*0
Formula (3) is the one given hy J. Arthur Harris, which is direct, hut not suited so
well to. machine calculation as (2) or (h) .
The computation may he illustrated with these data pn the length of corn ears in
centimeters and their weight in ounces.
\^ (s'x) 2 » S* 2  Nx 2 and (s' y ) 2 = Sy_
. SP „i£ . Nf c
H N  .
r = i P(x  x)(y  y) = ' i^ gxy _ xg.ix), iS(xj ±JULX
w s, x s 'v /lliE  sT) TIHZZjS 3 )
»' 1/fo" (Sxy  N x y  Ef;'x y + N x y } = Sxy  H x y
l'/w V Sx 2  Hx 2 ) (3y 2  Hy 2 ) V fex 2  Nx 2 ) (Sy 2  Hy 2 )
91
Length (x)
vei^rt (y)
X 2
y 2
xy
2.5
3.5
6.25
12.25
8.75
2.5
3.0
6.25
9.00
7.50
3.0
 5.0
9.00
25.00
15.00
4.0
T.O
16.00
49.00
28.00
*.5
55
20.25
30.25
24.75
5.0
8.0
25.00
64.00
40.00
5.5
8.0
30.25
64.00
44.00
6.0
10.0
36.00
100.00
60.00
6.0
7.0
56.00
49.00
42.00
6.5
10.5
142.25
107.62
68.25
6.5
6.5
42.25
42.25
42.25
7.5
10.0
56.25
100.00
75.00
8.0
8.0
64. 00
64.00
64.00
8.0
10.0
64. 00
100.00
80.00
8.0
12.0
64.00
144.00
96.00
8.5
13.0
72.25
169.00
110.50
9.0
12.0
81.00
144.00
108.00
9.0
14.0
81.00
196.OO
126.00
9.5
13.0
90.25
169.OO
125.50
10.5
14.0
107.62
I96.OO
147 .00
S(x) =130.5
S(y) = 180.0
s( X 2) = 949.87 S(y2)
= 1854.37 Sfor)=
1310.50
x = 6.5 y = 90
The symbols x and y are the means of the x and y arrays. The values, S(x^) and S(y2),
are the squared values for each separate entry of x and y, respectively, and the
summation of the same. The value, S (xy), is the summation of the product of each
value of x "by the corresponding value of y. In practice, only the sums of the various
values are recorded in machine calculation.
The values may he substituted in (2) as follows:
r = S(xy)  § x y = 1310.50  (20) (6.5) (9.0)
7 (Sx2Nx2)(sy2  Uy2) . 7 (9^987  845. 00) (1834.37  1620.00)
= 1310'50  1 17000  140.50 = 0.937
J (104.87) ( 2 14 . 3 7 1 722480.9819
Those who use this icrmula for the computation of r are warned that a serious error
may be introduced by dropping decimals. The means should be carried out to twice the
number of decimal places as appear in the original data. The formulae given above are
particularly valuable when K is ^•■•pII, i.e., less than 50.
V. Calculation from a Ccrrelai:J.or Surfac e
The correlation coefficient rauy bs calculated from a correlation surface with the
deviations from the assumed means taken on an arbitrary scale. It is necessary to
apply corrections for the means, standard deviations, and class intervals. Fisher
(1934) has made a contribution, to simplicity in the mechanical computation of the
correlation coefficient, his nethodl "being used "below. The data are for the correla
tion of total grain weight (x) in grams and culm length (y) in centimeters in wheat
plants .
■*In the determination of standard deviations where Sheppard's Correction has "been
used, the uncorrected standard deviations should he used in computing r.
92
Table 1. Correlation Table for Grain Weight and .Culm Length in Wheat.
Cudm^
Gr. Wt.
9.5
29.5
^9.5
69.5
89.3
109.5
129.3
1^9.5
169.5
189.5
209 . 5
length
^
(y)
Y \.
. . . s.
X
2
3
k
5
6
. 7
8
9
10
11 f .J
62
1
i
2
3
67
2
1+
1
7
72
3
1
4
3
2
p
2
l)+
77
k
2
8
1+
2
2
18
82
5
2
7
h
7
8
5
1
5*
87
6
2
6
12
5
8
17
6
1
1+ 61
92
7
8
23
2
2
16'
20
5
2
83
97
8
7
21
2k
l*+
3
1
1+
2
1 83
102
o
2
22
3h
13
1
1
8
6
1 88
107
10
1
3
32
26
6
T_
2
71
112
11
) +
15
1
•
26
117
12
5
1
6
f x
1
15
^3
9o
llif
90
61
35
21
12
6 I+.9I+
The data may be arranged as fellows
Table 2. Computation of the Correlation Coefficient
(1)
(2)
igth
(
culms
3) (*
) (3)
(6)
(7)
(8)
(9)
(10) (11
) (12)
(13)
(Ik)
Av . 1 61
(y)
Total
Prod
Ay. grain weight (x)
Total
Prod
for
. uct
for
uct
Class
Wt.
Class
length
Center
Y
%
Yf
Y 2 f
y
S 'Xf
YS'Xf
Center
X
fV XfV
x 2 f x
S'Yf
XS 'Yf
62
1
3
3
3
10
10
9.5
1
1 1
1
3
3
67
2
7
ll+
23
19
29.5
15 30
■ . 60
■ 51
102
72
3
11+
1+2
126
1+8
11+1+
1+9.5
3
1+3 129
387
254
762
77
k
18
72
288
lh
296
69 . 5
1+
96 38I+
1536
696
2731+
32
5
3^
170
850
167
835
89.5
3
114 570
2850
963
1+815
37
6
61
366
2196
363
2178
109.5
6
90 5I+0
3240
770
1+620
92
1
83
581
1+067
[ i93
3I+65
129.3
7
61 1+27
2989
1+66
3262
97
8
83
661+
5312
1+33
3621+
11+9.5
3
33 280
22^0
2 1+6
1968
102
9
88
792
7128.
500
1+500
169.5
9
21 189
I70I
178
1602
107
10
71
■710
7100
1+Q2
1+020
189 . 5
10
12 12.0
1200
101+
loi+o
112
11
26
286
31I+6
161
1771
209.5
11
6 66
726
1+1
451
117
12
6
72
861+
1+1+
323
Totals
h 9 k
3172....
11108
27J6
21^0o_
494 2756
1 69JO
3772
211+09
,qy
ay£
P5YV
q Tr
  2~
rty
Y  377f
49I+'
 7.63%
X  27J6
1+01+
5.538^
The details of computation are explained as follow
1. To simplify the arithmetic,, the variables X and Y are used in place of x and y,
respectively. They are related by: . ,
93
X = (x  x x ) /C x + 1
Y = (y  y^/Cy * 1
where X]_ and y}_ represent the class centers of the first classes, and C x > Cy, are
the class intervals of the x and y distributions, respectively.
2. The values for Yf y (in column k) are the products of the class values, Y, and their
respective frequencies. The values for column 11 are computed in a similar manner,
3. The values for Y^fy (in column 5) are the products of columns 2 and h for the
respective values of Y. The X^f x values in column 12 are computed from columns
9 and 11.
h. The total deviations in culm length (y variable) are shown in column 13 for each
column for grain weight (x variable). Here the symbol (f) without subscripts
indicates the frequency of one cell of the correlation table, i.e., the frequency
of a particular value of X accompanied by a particular value of Y. The symbol S'
denotes the total over ,Just one array. It is necessary to refer to Table 1
(columns. 2 and 3) bo compute these values.
1st Yarray = (l)(3) =3
2nd Yarray = (l)(l) + (10(2) + 00(5) + WW * (2)<5) + (2)(6) = 51
3rd Yarray . (1)(2) + (3>(3) + WW + (5)(7) ♦ (6) (6) + (7)(8)
+ (8)(7) + (9)(2) + (10)(1) = 25^ etc.
The values for the Xarrays in column 6 are computed in a similar manner.
5. For the product (XS'Yf) multiply each value of the total for length in column 13
by its respective Xvalue in column 9. For example, (3)(l)  3, (51) (2) = 102,
etc. The values in column 7 are computed similarly. It is noted that the ultimate
result (SXY) of the computations carried out in columns (6) (7) and (l3)(lU) is the
same. Thus, one provides a check on the other.
6. The computed values in Table 2 are then substituted in formula No, 2 above: \y
r = S(XY) NXY _ = 211+09  (1*9*0 (7 6356K5. 5585)
V (SX2  hx2)(sy2 NY2) j [3.6930  (k$k) (55385) 2 J [31108(i+9 J +)(7".6356F]
= 21,1+0?  26,891.16 , 3rjt& t
7(16,930  15,155.^5)(51,108  28,801.39) J 4,097,808.0 0""
= 517. 81+ / 202I+.30 = 0.2558
^The data in the problem above have been coded. Suppose a = assumed mean, and C =
class interval. It can be shown that the correlation coefficient from coded data
is equal to that from the natural numbers, viz., r xy = r^y.
x = (X  a x )/C x and y = (Y  a y )/C y
x  x = C X (X  X), and s' x = C x s' x
= 5(x  x)(y  y) = C x C y S (X  X)(Y  Y) e
s 'x s 'y c x C y S 'X B, Y
9^
7. The true means can "be computed from the above values as follows:
y = (Y  1)G 1 +Tft x (7.6356  1)(5) + 62 = 95.178O . ;
x =» ■ (X  1)C X + X ± ■ « .(5.5385, 1)(20) + 9.5  100.2700
71 . Use of the Corr elation Coe f ficient for Error of a Dif f erence
The correlation coefficient may he used to reduce the standard error of a difference
(o~cl) when there exists a correlation "between the paired values of two variables.
This usually enables one to obtajnaignif icance with smaller differences than is pos
sible with the formula, o~ d = J a + b , given previously (See Chapter 6). However,
it is seldom worthwhile to apply the correlation formula unless "r" is large because
the reduction in error is usually insufficient to justify the greater amount of cal
culation. The extended' formula for the standard error of a difference is as follows:
o" d
= y a 2 + b 2  2 r al3 ab     (5)
In this formula a and b represent the standard errors of the separate values being
compared, and r, the coefficient of correlation between the separate measurements of
these quantities.
The averages for the heading and blossoming stages of irrigation of spring wheat over
a 9 year period v may ■ be taken to show the value of the correlation coefficient in
the reduction of the standard error. The average yields of grain in pounds per plot,
together with their standard errors (e>), are as follows: ' '■
Stage of Irrigation
Year Heading B lossom ing
The coefficient of correlation
was calculated for the paired .
annual yields by the use of
the formula for the ungrouped
data as explained in paragraph
IV, viz., r = + 0.^06.
1921
^78
^52
h^o i h6
1922
776
* 57
637 * 31
1923
lllk
± 58
9U7 ± h9
1921+
1218
* 53
1189 i 52
1925
555
t 28
524 ■* 27
1926
llh
* 59
6J+5  k?
1927
l(&3
59
1035  39
1928
639
* ^
6lk ± 35
1929
895
113
839 ±ro7
Mean
333
± 19
762 ± 18
°7) » V
a 2 + b 2
" 2 ^a
b ab
= ^9) 2 +
d/a d =
7l/20.r
f
3.52.
(18) 2  (0.812)(19)(18) = 20.17
The standard error of the difference, calculated without the use of the correlation
coefficient to reduce 'che error, was as follows:
o d = Va 2 + b 2 = J(l9) 2 + ( l8 ) 2  2 ^.1T
d/o d = 71/26.17 = 2.71
vY] ss first true class value. C Y , C v  class intervals.
v^P.obertscn, D. W., et al . Studies on the Critical Period of Applying Water to wheat
Data from Colorado Experiment Station,
95
It ie apparent how the test comparing the averages of the yearly means is strengthened
"by taking into account the correlation due to years,
VII. Significance of the Correlation Coefficient
The test for significance is to determine the probability (P) that the observed
correlation could have arisen by random sampling from a population in which the corre,.
lation is zero. The ttest is more accurate for small samples while the standard
error test is satisfactory for large samples.
(a) The Standard Error Test
In large samples drawn from a population in which the mean value of r is
zero, the standard error of "r" is given by:
a r =
1 jg. (6)
Vff  1
From the standard error, r/o r is computed to determine significance. When r/or is •
less than 2.0, the relation is probably due to chance rather than to correlation
between the variables compared. Fisher (193*0 states that, in the use of the above
test, the value of r itself introduces an error which is magnified when r is squared.
Only in the case of large samples (greater than 100 pairs of observations) can the
standard error test be used safely. Further, the distribution of r, at least for the
stronger values, is so skewed that it is unwise to make any interpretation of differ
ence in terms of cr r based on probabilities related to the normal curve.
(b) The ttest for Significance
For small samples, the distribution of r is not sufficiently close to normal
to justify the ordinary standard error test. Fisher (193*0 has developed the "t M '
test as a more accurate test for significance. Thin test measures the probability of
obtaining a given value of r from a sample of paired values of a given size due to
chance alone. A value of this probability of less than P = 0.05 indicates that the
association of the characters is not due to chance, therefore being significant. The
formula for "t" for a correlation coefficient is as follows:
t = r Vl^"2~ (7)
7 1  r 2
In this formula K = the number of pairs of observations. The degrees of freedom for
the estimation of a correlation coefficient are N  2 due to the fact that two statis
tics are calculated from the sample.
The use of "t" may be illustrated with the correlation of ear length (x) and weight
(y) in corn (Par. IV).
t = rVN  2 = 0.937 /gQdL  = 11.38
V 1  r2~ V 1  (0.937)*'
In the "t" table it is noted that for 18 degrees of freedom, the value of t required
for P ■ 0.05 is 2.101. Thus, the above value is judged to be highly significant.
The same result can be obtained from Table VA in Fisher, (193*0 .
(°) Difference between Correlation Coefficien ts
A test for the significance of differences between correlation coefficients
*tf» b'ien. suggested by Fisher (193*0 as follows:
z' = 1/2 [lege U + r)  log e (1r)] ,   (8)
96
The standard, error would "be as follows
o z , = 1/VN  3 ~ . .:.____,,,__ (9)
The method may he illustrated from an example given "by Goulden (1937) who studied the
relation "between the carotene content of wheat flour and the color of bread for 139
wheat varieties. , The. correlation. coefficients were as follows:
Carotene in whole wheat with crumb color", ri = 0.^951 ■ ' •
Carotene in flour with crumb color, rp =' 0.5791 •
The z' test would be applied as follows:
z 'l = 1/2 [logg (1 + 0.1+95D  log e (1 ~ 0.1+951)]
= 1/2 [log© 1.^951  log e O.5049]
a 1/2 lege 2.9612 == 0.5^28
= 1/2 log
t>e
1.4951
0.50^9
• 2 = 1/2 [log e (1 + 0.5791)  log e (1  0.5791)]
/2 log
I4I91 I „ 1/2 log 3.7517 = 0.6612
0A209 I ', ee •
5 A = z{  0.6612  0.5428 = o.llSU
°k'2  zN = /I + J  0.1213
1 V 136 136
dz'/o z i = 0.1184/0.1213 = 0.9761
Since the difference is less than its standard error, it is not significant.
The formula for z' deals only with the numerical value of r, no attention being paid
to algebraic signs. It may be noted that the z test for significance of r is superior
to the devices heretofore described.
VIII. Interpretation of the Correlation Coefficient
Certain precautions are necessary in correlation analysis. First of all, the charac
ters of the individuals under consideration must be paired for some logical reason.
The sample should also be representative of the population. Ordinarily it is inad
visable to calculate correlations on numbers where N Is less than 30. Caution should
be used in the application of correlation statistics where I is less than 50.
Spurious correlation is a condition where the things compared are not causally relat
ed, hut which are related to a third cause. ■ Frequently there Is a tendency to assume
that a significant correlation coefficient is proof of a causal relation between two
variables. This may not be true. Extreme caution should he used in inferring cause
from a correlation coefficient .
S  Linear R egres sion
IX . Th eory of Regre ssion
A regression is said to be linear when the means of the sets of values of one charac
ter which correspond to given values of the other character can be well fitted gra
phically by a straight line. Under such conditions the coefficient of correlation (r)
97
is a valid measure of association. From the definition, it is evident that there
must he two regression lines. They are termed the lines of regression of x on y, and
y on x.
ydistri
hution
xdistribution
A Jc
X
■ ■  , 
*\* V
\ 1
: \. )
B
This diagram should give a clear con
ception of what is meant hy regression.
The elliptical nature of the scatter
is shown with the dots which indicate
the means of the individuals in each
array, fcoth horizontal and vertical.
The means of all the rows fall approxi
mately in a straight line, as well as
y those for columns. These lines, called
the regression lines, intersect at a
D point which indicates the means of the
two general distributions, x and y.
The mathematical equations of these
lines can he obtained by the method of
least squares. The line AB is the
regression of x on y. Its equation is
as follows:
X  x 
s x (y  y)
H
(9)
Where x is the value estimated, say x e .
The line CD is the regression line of y on x. Its equation is as follows:
y  y
= r _fy_ (x  x)
(10)
Where y in the val\ie estimated, say y e .
These equations may be used to predict or estimate the most probable value of one
character to accompany or be associated with a given value of the other character.
When a certain value is given y in the equation x e " x = s' x /s'y (y  y) , one can
solve for the predicted value of x that corresponds to it. Likewise when a value is
given to x, in y e  y = s'y/s' x (x  t) , the most likely value for the y that accom
panies it can be found. Predicted values given by the regression equations are con
servative. Actually, the term "regression" is a result of this tendency. These
equations have little or no value for prognosis unless r is quite strong. The two
diagrams below depict predicted values given by regression equations in two case 3,
i.e., where the correlation is strong and where it is weak.
A x — »x
Cane I
93
In each case, observe the same given value of x indicated "by the point 5c, on the
upper line of each diagram. The vertical line from' the point, x, to the line CD
measures the predicted value of y. The portion (A B) illustrates the amount .of" ab
normality predicted. This is seen to "be much smaller in case II where the correla
tion is weak. Moreover, the standard error of an estimated value is so large that,
unless "r" is high, the reliability of an estimated value is small. Although a sin
gle predicted value is of little avail unless a very high degree of association exists
between two characters, the regression measured by the coefficients r b^ / e£ and
r a j / si ^sy be quite appreciable i.n one case or the other, even when r is small.
This is due to the fact that variation in one character may be quite low. For in
stance, the association between the yield of a crop obtained from several plots and a
certain treatment given in various degrees of intensity to the plots may be quite
low. The first reaction would, be that the treatment is not justified. However, the
regression might be appreciable, so that the treatment might be very worth while for
the crop as a whole .
The more important Interpretation of a px"edicted value from a regression equation
lies in the fact that it may be considered as a mean estimated value of the variable
which may be expected to result in connection with a number' of identical values of
the second variable. Such an estimated mean would have a standard error = js^ where
m = number of repeated cases of the second variable, and s e is the standard error of
regression (See Section! (c) below).
X. Computation of Regression Equations for Grouped Data
The equation for the regression coefficient is as follows:
b VT = SY(X  1)
<y " S (X  X.)'<
The most convenient formulae for machine computation are as follows
VT = ox l^.._.r_AJ ____ _ _ _' . ________ _____( 1 1)
S (X X.)2 U±j
. s(x 2 )  (sx)7n ," " " ' '" " K ~ '
or
V = TIS(X Y)  (SX )(SY) _ (13)
NS(X2)  (SX)2
Where the transformed variables (X,Y) are "not used, the same relations held in terms
of the original variables (x, y) .
i • '
( a ) Comput ation of Regres sion Coeffi clents
The computation may be illustrated for the correlation between total grain
weight and average lengths of culms in wheat plants (Paragraph V above) . Calculations
from Table 2, which can be used here, are as follows for coded data:
SXY = 21,1+09 SX = 2736 SY = 3772
X a 5.538e 1 = 7.6356
N  I+9I+ SX 2 = 16,930 SY 2 = 31,108
By substitution in formula 13:
b vx = JBlXYl^lSXKSYL = (W (2H109) ~ (2736) (3772)
JX m (X2)  (pj)2 (W) (16, 950)  (2736)2
= 10, 576,01+6  10,320,192 = 2^5,851+ = 0.2913
8,363,1+20  7, ] +35,696 877,721+
99
D = M5(XY)  (SX)(SY) = (I^IQ (21^09)  (2756)(3772)
BS (Y*)  (SY) a (W( 31108)  (3772) 2
= 255, 85^ , 255, 85^ = 0.2246
15,367,352  lk,227,9Qk 1,139,368
(h) Substitution in Regression Equation
The equation for the regression of Y on X is as follows:
Y e = I  b^ i + h yx X
= 7.6356  (0.2915) (5.5385) + (0.2915) x
= 6.0211 '0.2915X^/
The equation for the regression of X on Y is calculated in a similar manner.
X@ = X  hxy Y ' bxy Y
= 5.5385  (0.2246) (7.6356) + (0.2246) Y
= 3.8235 + 0.2246 Y
(c) Significance of Regression Coefficients
The "t" test for significance of the regression coefficient, h yx = 0.2915
(coded "basis) can he determined as follows:
S (Y  Y e ? = S (Y  Y) 2  h 2 x S (X  X) 2
= SY 2  Nf 2  h 2 x (SX 2  H£ 2 )
= 31,108  (494)(7.6356) 2  (0.2915) 2 [ 16,930  (W(5o385) 2 ]
= 31,108  28,801.3856  0.0850 (16,930  15,153.^500)
= 2,306.6144  (O.0850) (1,776.55) = 2155.6076
s e =
S(Y  Y) 2  b 2 S(X  X) 2 = /2155.6076 = 2.0932
F~rp v ^92
byy j S (X  X) 2 = 0.2915 7l776.55 = 5.87
s e 2.0932
This indicates that the regression coefficient is highly significant. The coefficient,
t, x = 0.2246, can he tested in a similar manner.
■"The coded values are changed into actual values hy the conversion of Y to y, X to x,
and the multiplication of hy X hy C„/C x as follows:
y = (Y  1) C y + y x = (7.6356  1)(5) + 62 = 951780
x = (X  1) C x + x x = (5.5385  1)(20) + 9.5 = 100.2700
h yx = (0.2915) (5/20) = 0.0729
y e = y  h yx x + h yx x = 95.1780  (0.0729) (100.27) + (0.0729)x
= 87.8685 + 0.0729x.
100
1. Eldertcn, W. P. Frequency Curves and Correlation. Lay ton, I906.J
2. Ezekiel, M. Methods of Correlation Analysis. John Wiley' & Sons. ■ 1930.
3. Fisher, R, A. Statistical .Methods for Research Workers (dth edition) Oliver
and Boyd. pp. I0OI97. 193k. • ..• ;
k. Gouldon, C. H. Methods of Statistical Analysis. Wiley, pp. 5277. 1939.
5. Hayes, E. K., and Garber, R, J. Breeding Crop Plants. McGrawHill. pp. il3T>5<
1927.
6. Snedecor, G. W. Statistical Methods. Collegiate Press, pp. 89133. 1937
7. Tippett, L. II. C. The Methods cf Statistics (2nd edition). Williams and
Worgate. pp. l4ol88. 1937.
8. Treloar, A. E. Outlines of Biometric Analysis. Burgess, pp. UO63. 1933'
9. Wallace, H. A., and Snedecor, ,G. W. Correlation and Machine Calculation.
Collegiate Press (Ames) . 1931.
Quest i oris f or_ Bis cue si on
1. Define correlation.
2. What is a scatter diagram? How is it influenced by high correlation? Low
correlation?
3. When are two variables said to be correlated? riot correlated?
k. What is the generally accepted method for the measurement of correlation? Its
limitations?
3. What is meant by r = + 1, r =  1, and r = 0?
6. How can the standard error of the difference bo reduced by the use of correlation';
7. Why is the "t" test preferable to the standard error test for testing the signi
ficance of r?
8. What precautions must be exercised in the interpretation of the correlation
coefficient? Why?
9. Under what conditions is "r"'a valid measure of paired relationships?
10. What is regression? Its use?
11. Explain what is meant by the regression of y on x. Regression of x on y.
Problems': ..
1. These data were collected to study the relationship between the soil moisture con
tent and the yield of wheat (Data from Salmon) :
Mo i st ure ( x ) Y i e 1 d ( y ) Mo i st ur e ( x ) Yield (y) Meisture(x) Yleld(y) Moist ure (x) Yleld (y)
21 1 25 3d 18 10 le 3
17 1 23 • 2k 21 28 15 k
17 3 26 39 21 28 18 8
18 3 18 22 25 17 12
21 21 18 .0 23 29 17 13
20 2k 18 17 17 16
19 20 18 16 7 16 15
19 7 18 3 15 o 19 11
17 19 19 7 > 19 9 19 10
16 21 15 11 18 23 18 k
16 21 15 9 18 23 21 k
16 20 13 9 18 27  13 36
2^ 32 15 9 18 23 26 k'J
2k 37 18 13 jl° 3
(a) Calculate the coefficient of correlation (r) by the machine method for ungrouped
data, (b) Test the significance of "r" by the "t" test.
101
2. The average length of culms and the average diameter of culms was measured on k$6
wheat plants at the Colorado Experiment Station. The data follows:
Av.
Diameter
culms
(mm. )
Avera
ge length of
culms (cm.)
: x
60
65
70
75
80
85
90
95
100
105
110
115
y
6k
69
7^
79
8k
89
9k
99
10U
109
llU
119
91
100
1
1
101
110
1
2
111
120
1
1
1
121
130
1
k
2
1
2
1
131
1^0
1
6
3
3
2
1141
150
2
3
1
6
lv
2
1
151
160
2
5
5
16
27
8
7
1
161.
170
1
3
8
13
23
25
26
6
2
1
171
180
3
1
9
10
15
26
13
8
8
181 •
190
1
9
10
10
23
32
1+
3
191
200
2
2
8
13
20
12
1
201
210
2
2
3
5
k
1
211
220
2
Calculate r and test it for significance with "t" test.
3. The correlation between the reaction to Helminthosporium in F3 and F5 barley lines
was studied at the Minnesota Experiment Station. The reactions are given in per
centage infection for 1921 and 1922. The data follow:
Percentage in 1921
Percentage
in
1922
12
15
13
21
2k
27
9
12
15
18
21
2
2
2
1
1
3
2
3
1
1
2
3
2
1
1
1
k
2
1
1
Calculate the coefficient of correlation and the regression lines,
regression lines.
Plot the
k. The 9year average yields of wheat for the period 192129 were as follows when
irrigated at the germination and filling stages. Five plots were averaged each
year. The data follow in grams per plot:
102
Year
Grown
Genuine
it Ion
(o S )
192.1 '
511 ±
*7
1922
655 *
21
1925
91A t
52
1924
952 t
23
1925
1+70 ±
16
1926
557 
29
1927
783 ±
20
1928
. r ■" J r
066 
20
1929
756 ±
65
Filling
(gra.)(c£)
•321
23
518
~
17
733
i
26
125
4.
3k
538
±
18
601
f 
31
812
i
511
•f
18
733
z
63
Means  685 ± 11 6'57 * 11
Determine whether or not the wheat irrigated at germination differs significantly
in yield from that irrigated at /joint in& and tillering. Calculate o c ] of an average
of a difference "by the formula 07,  J a^ ■*• h 1 ^ , and by the extended formula for use;
of r.
Compare d/a& for both formulas.
CHAPTER X
THE ANALYSIS OF VARIANCE
I. Generalized Standard Error Methods
The "basis and purpose of all statistical methods is to analyze and measure variabili
ty. Variation between observations may be due to one or more recognizable causitive
factors. In addition, in all statistical work, there occur variations between obser
vations that result from the coalition of a large aggregate of chance factors which
defy control. This latter type of variation between observations results in various
types of statistical distributions when it is attempted to describe homogeneous popu
lations.
Suppose one considers a population wherein the variability may be due to both the
combinations of innumerable chance factors and the nonhomogeneity of the population.
In other words, the population naturally and logically submits to subdivision into
several homogeneous groups or sub populations. Such a situation is common in variety
tests in field experimentation. Generalized standard error methods have been devised
for data of this kind.
The purpose of the generalized standard error methods is to compute the standard error
of an entire experiment in order to increase the accuracy of the estimate of error.
In a variety test where each variety or treatment is replicated, say four times, the
reliability of the results would be very low were one to compute the standard error
for each variety separately. However, the estimate of error would be much more re
liable when computed on 10 different varieties, each replicated say four times, in
the same experiment. In this case a total of hO plots would contribute to the esti
mate of error instead of four.
The analysis of variance, developed by B. A. Fisher, has proved to be the most precise,
flexible, and readily usable method available for the analysis of the results from
field and many other biological experiments. It consists essentially in the partit ion
and apportionment of the total variation tc lbs known causes with a residual portion
ascribable to unknown <:■:" uncontrolled variation and therefore called experimental
err or. When the variability is measured in suitable terms, i.e., sums of squares of
deviations about the means, the variability ascribed to the various causes will be
strictly additive. The calculations are therefore extremely simple. The mean value
of the sums of squares (mean square, variance, or standard error squared) is found by
division of the sums of squares by the appropriate number of degrees of freedom.
The literature on the analysis of variance has become very extensive during the past
15 years. It was first set forth in its complete form by Fisher and MacKenzie in
1923 For its application to field experiments, Fisher (193*0 } and Fisher and Vis
hart (1930) have given excellent discussions. Among other sources of information on
the application of the analysis of variance to field experiments may be mentioned the
books by Snedecor (193^ and 1937); end Tippett (1937), and papers by Eden and Fi3her
(1929), Goulien (1931), Immer, et al (193*0, and Wishart, (1931). For a summary of
the mathematical theorems involved in the analysis of variance, the work of Irwin
(1931) is recommended.
A — One Criterion of Classifi cation
H. Theory of First Special Case
Suppose a sample is formed from the general population with random samples of equal
size taken from each of the eub populations. In case m subsamples contain n measure
ments each, the total sample will contain N = nm measurements. It is how proposed to
103
104
analyzc the total variance, I.e..,
^ = sj_l__j___;.  .. .   — —  .   . (i)
N  1 .■■.;.
where x Is the mean of the total sample. Let x represent an individual measure of
any (i r th). : sub sample. Then,
x  x a (x  %) + ( X1  x)   _ _ _ .. . . (2)
where £]_ is the mean of the ith sample.
First, the above identity should be squared and suxamed for, all the n individuals which
form the ith subsample. The symbol 8' will be used for this summation.
S' (x  £..)*£ S' (x  Xi ) 2 + 2(%  x) S'(x  %) + nCxj  x) 2     (3)
Since S'(x  x^ ) = 0, it is evident that the second term on the right vanish
es.
m
Now suvvpose one simis over all the m different sub groups by use of the. symbol, S.
TYJ _ Oil' ^ J J
The combination., SB' is simply S, or summation for all individuals of the total i sam
' 1
Pis
m . v o la , ' . o m . _ ' .0
S (x  x) = SS« (x  ±) d = SS • (x  %) + n S (x 3  x)'      (k)
The term on the left is the siim of the squares of the deviations of the individual
observations from the means of the subsamples . The second term on the right is n
times the sum of squares of the deviations of the means of the subsamples from the
mean of the total sample.
The computation of these three terms is most easily accomplished as follows:
(1) Compute the term on the loft,
S(x  x) 2 „ Sx^  (Sx) 2   _..__.__. _ ~ (5)
N
(2) Next, the second term on the right,
n S(%  x) 2 = n S S, 2  (So:) 2 = S(x a 2 }  (Sx) 2  (6)
1 1 N TT — r 
whore x a Is an abbreviation for S'(x), the total of the Variates in a single
sub sample .
(3) The other term may be found by mere subtraction.
The difficult thing to explain comes at this point. It would be easy to merely
apportion, for the sample in question, the total variance into the variance within
subsamples and into that between sub samples. Those two respective variances could
be obtained by division of the first and second terms on the right of the identity b,y
N. However, the real desire is to obtain the best estimate to the variance of the
population as exhibited within the sub populat ions on the one hand, and betw een the
sub p op ulat ions on the other.
The best estimate of the total variance of the general population is given by :'
s 2 * = SLx^ x) 2 ......   ... (T)
N  1
Where K  1 (or nm  I) is the number of decrees of freedom.
Likewise, the best estimate of the variance within cub populat Ions (replicates
agronomically) will be:
O xl m. . „ ,0 • /n\
V 1 __ ~_ 1 '
' if m"
NJ/ "is estimated by" in this sense
105
Where N  m = m(n . 1) la the number of degrees of freedom. This is true "because m
separate means of the m subsamples were used in the computation.
This expression, s 2 r = the variance within subsamples, is often called the residual
variance .
The last term, n 5 (x^ x) 2 , (Equation No. k above), must now he considered.
At first glance, it would seem that the sum of squares of the deviations of the means
of the sub samples from the mean of the total sample when divided by m  1, the number
of degrees of freedom, would give a proper estimate of the variance between subpopu
lations. However, xj_ does not represent the mean of the ith sub population, but
rather the mean of the ith subsample. Therefore, the difference, x± x, is due to
the combination of (1) the inherent nature of the ith sub population and (2) sampling
fluctuations within the ith subsample. Thus, n S (xj  x) 2 can be interpreted to
m  1
estimate the sum of ns^ 2 , n times the variance of the means of the sub populations,
and s r , the variance within sub populat ions .
It should be remarked that the expression,
s r 2 = SS ' (x  Xj)^, called the "variance within sub samples", is simply
N  m
one estimate of the variance of the total population.
m _ «2 ? o
The term, n S (xj  x) = n s£ + s£, called the "variance between sub samples", is
1 m  1
n times the variance of the subsample (treatment) means about the total sample mean
with st, added. On the assumption (null hypothesis) that the true variance of the sub
population means about the total population mean is zero, it then becomes clear that
"variance within subsamples" and "variance between subsamples" are both independent
estimates of the same concept, i.e., variance of the total population.
The material may be placed in tabular form for clarity:
Source of Sums of
Variation Squares
m p
Between Subsamples nS(x i  x)
m  o
"Within sub samples SEP (x  x^)
Total S(x  x) 2
Degrees
Freedom
of
Estimated Mean
Variances
m  1
n B t
2 + a /~
~H 2
N  m
°r 2
• V
N  1
8 a
= v 2
The first two entries in the last ccl'^ian may be examined, i.e., the estimates given
by the sample. These are nsj2 + B J£ . it is obvious that the fjrst should exceed the
second unless s^ 2 is zero. >2/
It is now desired to determine whether or not there is a significant variation be
tween the sub populat ion, i.o.., whether c^ 2 is significantly different from zero. An
estimate of o^ 2 may be made by subtraction of the estimate of cr r 2 from that of nc=t 2 
This result is then divided by n.
^Occasionally the reverse is true. This apparent contradiction is explained by the
fact that the results are merely estimates which may be distorted to whatever extent
sampling fluctuations may account .
106
So far it is obvious that tho first step in the analysis of variance serves two pur
poses: (l) It gives a method to test the homogeneity of a population; (2) It gives
a convenient method to test tho dif f erencos "between several meant; as a whole.
Probably the best method to test for association between sub populations is through
the use of the "z" index devised ~by Fisher (193*0 • This affords a test of signifi
cance between two variances, e. g*, &y~ mid $2 
7. ■■'. i los Bi' 11   1 1 act Firs' ■ « 1 O'.v ,qi' ___..__.__.._._,__ ( O)
— p
t.
The "z." table devised, by Fisher (193^0 niay ho used to test these values for signifi
cance through use of the number of degrees of freedom pertinent to each computed
variance. In this case, noj. + erg takes the place of sg; while cr r c takes tho
place of So".
Tests of significance may also be made Xrj means of the "F" test derived by Mahalono
bis (1952) and by Snedecor (193*0 ( 1937 ) • The table by Snedecor is the more extensive
The value "F" is the quotient obtained by division of the larger ^y the smaller var
iance. The "F" and "z" tests arc; equivalent since z =  loggF.
III. Computation for Single Criterion of Classification
This case may be illustrated with some data for the yields of two barley varieties,
(See Chapter o) . The yields in bushels per acre for the Glabron and 'Velvet varieties
grown in single plots on 12 Minnesota farms were as follows: (Data from F. K. Immer)
Velvet ( ro ) Total
Farm No.
Glabron (xi)
1
IlO,
2
kl
3
39
11
37
5
k6
52
7
51
8
37
Q
k?
10
k?
11
'46
12
6k
Totals (Sx)
3o
Means (x)
■'+3.
3333
S ( X 2) r, p 0]
399 . 00
kl
7 r *
30
32
kl
Vi
)+o
36
k2
39
k'(
30
309
V2A167
(§x)2 
k 9 .
k 13 .38
N
91
0i.
i /
60
87
93
Q6
113
37
Sk
93
10 : ;
1089
Suppose that the total variability is separated into two components, viz., that "due
to varieties" and that due to variation between plots of the ssme variety. The ex
pression "due to varieties" simply means that it is proposed to make varieties the
criterion for the break down, of the total sample into subsamples.
The sum of squares for total variation is found by summation of the squares of the 21 1
plot yields, e.g. ('+9) ;  + (i+7) + J  (.39) = 30,599? and the subtraction of
the correction factor (SxV/N from this value.
107
Algebraically, this is given "by:
Total ssS^fx  X) 2 = S(x 2 )  (Sx) 2 /N = 50,59900  49,413.38 = 1185.62
The sum of squares for. varieties is obtained "by the summation of the squares for the
two totals for varieties, dividing "by number of plots or values contained in each
variety total, and subtracting the correction factor, e.g.
Between Varieties = n£> (x^  x) 2 = SXy 2  (Sx) 2
1 "~n~ T~
= 595, 481. 00  49,413.38
12
= 4Q,623.42  49,413.38 = 210.04
The sum cf squares for within varieties, here used as error, is the remainder after
subtraction of the sums of squares for varieties from the total, e.g. H.85.62 —
210.04 = 975.58.
The analysis of variance follows:
Variation
due to
D.F.
Sum
Squares
Mean
Square
Standard
Error(s)
z value
obtained 5 pet . point F
Between Varieties
Within Varieties
1
22
210.04
975.58
210.0400
44.3446
6.6592
0.7777 0.72Q4 4.737
Total 23 1185.62
The degrees of freedom for "between varieties" and "total 11 are one less than the
number of varieties and total number of plots, respectively. The degrees of freedom
for within varieties are those for a single variety (11 in this case) multiplied by
the number of varieties (2 in this case), or (2) (11) = 22.
The mean squares are obtained by division of the sums of squares by their respective
degrees of freedom. The standard error of a single determination (s) is the square
root of the mean square for error (or variance) .
The ztest may be used to test the significance for variance "between varieties" and
that "within varieties". The value, z, is l/2 log e of the difference of the variances
to be compared. The values of the logarithms needed in computing z are found in a
table of natural logarithms (See Table 4, appendix).
V = 1/2 log e 210.04  1/2 log e 44.3446
= 1/2 loge f 210.0400 \= 1/2 log 4.737 * 0.7777
N^S (x  x) 2 = Sx 2  2x S(x) + Nx 2
= Sx2  2x«Nx rf Nx 2
= Sx2  Nx2
Since Nx = Sjfx)
Sx 2  Nic 2 = Sx 2  S(x)x = Sx 2  (Sx) 2 /U
The decimal point may bo moved to the left on the mean square values to shorten
the work, so long as the resultant numbers arc greater than 1.0. The true log e
values will not be obtained but tho difference of z value is unaffected. A shift
cf decimals is particularly desirable, when any of tho mean squares are less than
1.0 to avoid taking a negative log,* .
108
The theoretical z value .is looked up in the table given, by Fisher . (193M* where N] in
the number of degrees of freedom for the larger : and Njb '^ ie degrees of freedom for the
smaller variance. In this case z  0.729'+ for the theoretical value for the j per
cent point. However, the interpretation is made more easily by using the "F" value.
The "F" value is the quotient of the larger by the smaller variance, e,g., F = 210. 0U/
kk.jkkS = k.'jk. In Snedeeor's table (Table 2, Appendix) for Nn  1 D»F, and N 2 =
22 D.F., it is found that the observed "F" lies between the J.O per cent and 1.0 per
cent points. The theoretical value for the S.O per sent point is U.JO. It may be
noted that F = f for one degree of freedom.
IV. The More General Case
Suppose that the number of observations in each subsample varies, and that they are
represented by n.] , n.g n,,. Then F = Sn± , The equation for the sample is as
follows:
S(x  %) d  SS'(x  x) c  = SS'(x  Xj) d •+ Sn;(xi  x)< : ' (10)
1 j. 1
m _ o
Again, SS (x  x^) divided by II  m, the degrees of freedom, will give the variance
within^ a group. However, it is now impossible to arrive at an estimate of s+g be
cause Si:u (x_  x)^ is affected by the different number of observations in each sub
1 ■ o Irl / \ o
sample. Therefore suppose that &*.<• is trulv aero so that S n : (x.  x)^ will estimate
,.J ■'■  ll  I X * J. '__
r • m  1
This assumption may be tested for the existence of an association between subpopu
lations. To do this, the valuer: SB' (x  xi) r ~ and £> ni(xi • xf are compared for
a significant difference. ' if  m 
In the field of agronomic experimentation this situation is rarely found because the
experiments are designed to permit a simpler setup for the computation of the statin
t i cal constant s .
B  Two or More Criteria of Classification
V . Theory of th e Exte nded Case of the Analysis of Variance .
Frequently, the complexity of the experiment that affords the data makes it necessary
to analyze the total variance into more than two parts in order to make the most of
the possibilities. First, reexamine the tabular arrangement for the first special
case, (Paragraph II) . Suppose the classification of the total population into sub
populations, which forms the basis of the above analysis, be termed classification
"A". Now suppose the total population lends itself to an independent classification,
"B". which contains "in' 1 ' classes. For simplicity, assume that the sample subdivides
evenly for this classification' with "n"' observations in each class. Thus, N = nm =
n'm' .
Previously, the heterogeneity in the total population for classification "A" was test
ed "oy a comparison of V + 2 with 1 2 .2 . It was necessary to tacitly assume that each
sub population for classification "A" was homogenous. Now, if the population submits
to a new classification "B", it is quite likely that the original sub populations were
not homogenous if classification "B' : has any logical basis. Lack of homogeneity in
the sub populations increases the variance therein. The residual variance, Vpg, may
be so affected in the comparison between Vjv and 'vie that the differences between
groups for classification "A''' may appear to be insignificant when the opposite is
true . ■: •.
109
Therefore, It is proposed to remove from the squared residual errors, SS ' (x  x^) 2 ,
the sum of the squared errors between groups for classification "B". ^This amount
will he termed n' a (x<  xjr, It represents m'  1 degrees of freedom, while x<
j=l J d
indicates the mean of the jth class of classification "B". The mean variance "between
groups for classification "B" will he designated as V^, 2 .
At first, one might expect the mean residual variance (Vp2) to be definitely reduced
in this manner, regardless of any justification for classification "B". This is not
true because the reduced sum of the residual squared errors now represents only
Nmm , + 1 degrees of freedom where N  ra degrees of freedom were represented before.
Thus, it is apparent that the new Vr 2 will not differ sensibly from its former value,
should the differences between groups be insignificant for classification "B". How
ever, the greater the significance of the differences between the groups for classi
fication "B", the more markedly Vp 2 will be reduced. Then the ratio V^/^will be
sensibly increased, with the result that the test for significance of differences
between groups for classification "A" is strengthened. The new tabular arrangement
of the analysis is as follows:
Source of
Variation
Between groups
(A)
Between groups
(B)
Residual
Total
Sum of Squares
n S (xi  x) 2
1
&' / *?
n' S (x*  x) c
1 J
S (x  x) 2
Degrees of
Freedom
m  1
m 1  1
N  m  m' + 1
K  1
Mean
Variance
V,
tr 2
f B
V 2 (or s 2 )
The entry for the sum of squares of the residual errors is left blank because, in
computation, it would be found by subtraction.
This process may be extended in the same manner to take into account other possible
classifications which might contribute to the heterogenous character of the original
population. The object is for the residual variance to represent variance due to
chance alone as nearly as possible. Furthermore, an increase in the scope of an ex
periment will proportionately increase, to within differences due to sampling fluc
tuations, all the sums of squared deviations incorporated into the analysis. Ilence,
V^ and Vt« will be increased proportionately since the number of degrees of free
dom they respectively represent are unchanged. The value V^ 2 will be increased to a
lesser extent due to the fact that the number of degrees of freedom represented will
be more than proportionately increased. Thus, V^/Vg and V^,/V E will be increased
which, together with the fact that a smaller z value is required to prove signifi
cance, make it more likely that positive conclusions can be drawn from the analysis
of variance.
VI. Computation for Two Criteria of Classification
The same data on the yields of Glabron and Velvet barley varieties are used to illus
trate this case. It is desired to determine whether or not there is a significant
variation from farm to farm as well as between varieties. Hence, the computations
will be for total variance, that due to farms, and that due to varieties. The resi
dual variance will be obtained by subtraction.
110
s(x  x) 2  s (x 2 )  (Sx) 2 /it = 50,599.00  1+9,1+13.33 = 1185.62
n ff(£ i  x) 2 = S(x v )  (SxT = 595,l!8l.OO  1+9,1+13.38 = 210.01+
• 1 n 12
n
« &(x.  x)2 = S(x f 2) _ (Sx)2 = 100, 269.00  49,1+13.38 = 721.12
1
N
2
The subscripts, v and f evidently indicate "varieties" and "farms".
The new tabular arrangement now becomes:
Variation
due to
D.F.
Sums
Squares
Mean
Square
Standard
Error (s)
F value
Farms
Varieties
Error
Total
11
1
11
721.12
210.0'+
25^ M
210.0^00
23.1327
1+.8096
9.08
23
1185.62
When the F table is consulted it is found that an Fvalue of 1+.81+ is required for the
5 per cent point. Thus, the added refinement through the removal of the variation
between farms greatly increased the significance of the difference due to varieties.
VII. Introduction to Analysis of Variance in Agrl cultural Exp eriment a
The principal difficulty to contend with in field experiments is the variation in
soil fertility over the area used in experimentation. The natural fertility usually
varies continuously. The art of planning an experiment lies in the arrangement of
the varieties, treatments or conditions under investigation in nearby plots. They
are usually placed within as small a land, area as is practically feasible. The entire
arrangement is then replicated over a larger area so that the variations caused by
regional changes in fertility may be removed from the comparison. The randomized
block arrangement, and its more restricted form, the latin square arrangement, are
commonly' used to make possible the removal of the general effect of soil heterogeneity
by means of the analysis of variance
iments 1 ' . )
(See Chapter en "Design of Simple Field Exper
In the use of the analysis of variance in field experiments it is assumed that the
distribution of the plot yields is normal. I.e., that it fits the normal curve. The
"agronomist is familiar with the fact that the variability between plots of the same
variety grown on land of high fertility is often less than between similar plots of
low fertility, The variability among plots of high fertility may be considered as
restricted by what may be termed "ceiling effect" which imparts an abnormal distribu
tion to the population. Fisher and others (1932) found evidence of negative skowness
in heights of barley plants selected at random from plots that received, various
nitrogen treatments. Eden and Yates (1935) obtained similar results with height
measurements of wheat plants. They made a practical test on these data to determine
whether the validity of the ztest would be destroyed by such nonnormal data. They
concluded that the ztest could be safely applied. .
Refere nces
1. Eden, T. and Fisher, R. A. Studies in Crop Variation VI. Experiments on the
Response of the potato to potash and nitrogen. Jour. Agr. Sci. 19:201213.
Ill
2. Eden, T. and Yates, F. On the validity of Fisher's Z test when applied to an
actual example of nonnormal data. Jour. Agr. Sci. 23:616. 1933 •
3. Fisher, R. A. Statistical methods for research workers. Oliver and Boyd,
Edinburgh, Ed. 5. pp. 199231. 193^.
k. Fisher, R. A., Immer, F. R., and Tcdin, Olof. The genetical interpretation of
statistics of the third degree in the study of quantitative inheritance.
Genetics 17:107.124. 1932.
5. Fisher, R. A. and MacKenzie, W. A. Studies in crop variation. Jour. Agr. Sci.
13:311320. 1923.
6. Fisher, R. A. and Wishart, J. The arrangement of field experiments and the sta
tistical reduction of the results. "Imperial Bureau of Soil Science, Tech.
Comm. No. 10. 1930.
7. Goulden, C. H. Modern methods of field experimentation. Sci. Agr. 11:681701.
1931.
8. Immer, F. R., Hayes, H. K., and Powers, LeRoy. Statistical determination of
barley varietal adaptation. Jour. Am. Soc. Agron. 26:403419. 1934.
9. Irwin, J. 0. Mathematical theorems involved in the analysis of variance. Jour.
Royal Stat. Soc. 94:284300. 1931.
10. Mahalanobis, P. C. Auxilliary tables for Fisher's Z test in analysis of variance,
Indian Jour. Agr. Sci. 2:679693. 1932.
11. Snedecor, George W. Calculation and interpretation of analysis of variance and
covariance. Collegiate Press, Inc., Ames ; Iowa. 193^ •
12. Snedecor, G. "W . Statistical Methods. Collegiate Press, \mes. pp. 171218.
1937.
13. Tippett, L. H. C. The Methods of Statistics. Williams and Norgate, London.
(2nd edition), pp. 125139. 1937".
14. Wishart, John. The analysis of variance illustrated in its application to a com
plex agricultural experiment on sugar beets. Wissenschaftliches Archiv fur
Landwirtechaft, 5:561584. 1931.
Questions for Discussion
1. What is meant by generalized standard error methods? Why are they useful in
agronomic experiments?
2 . What are the general features of the analysis of variance?
3. What is the basis of subdivision of the sample for one criterion of classifica
tion?
4 . Why is it logical to use the variance for within varieties to compare with that
between varieties?
5. What is the "z" test for significance? "F" test?
6. How may the subdivision of the total sample into two criteria of classification
strengthen the experiment?
7. What assumptions are made in the use of the analysis of variance for plot yield
data?
Problems
1. Yield data in bushels per acre for 5 wheat varieties are given on the following
page:
11:
Replications
Variety 12 3 Total
32.^ 3U.3 37.3 10^.0
■r £
B 20.2 27.5 25.9 73.6
c 29.2 27.3 30.2 87.2
D 12.8 12.3 ll:. 8 39.9
1 . 21.7 gl+,5 23 Jl  69.6
Totals 116.3 126.1+ 131.6 37I+.3
(a) Calculate the analysis of variance for one criterion of classification,
i.e., "between and within varieties. •
(b) Obtain the "!?" value and determine whether or not the varieties differ
significantly in yield.
(c.) Use the "?." test to determine significance.
2. Calculate the data in problem. 1 for 2 criteria of classification, i.e.., replicates
and varieties. Determine whether or not the varieties differ significantly in
yield by use of the ;; F" test.
.
CHAPTER XI
COVARIANCE WITH SPECIAL REFERENCE TO REGRESSION
I. Relationship of Covariance, Correlation, and Regression
The concepts of covariance, correlation, and regression are interwoven, being funda
mentally equivalent. Suppose one considers N pairs of measures that relate to two
characters represented "by the variables x and y. In the chapter on correlation, it
was seen that the basis for the measurement of correlation and regression was the
product sum, S(x  X)(y  y). The entire subject can well be treated by the analysis
of variance principle.
II. Analysis of Covariance
Suppose the sample of N pairs of measures is divided into m subsamples that contain
n pairs of variates each. Let x^ and yj_ represent the pair of means that correspond
to the ith subsample. Then, for any pair of variates in the ith subsample this
equation can be formed:
(x  x)(y  y) = [(x  Xi )  (%  x)] [(y  f ± ) + {f ±  y) j    (l)
By an analogous procedure to the first treatment of the analysis of variance, the
right side of this expression may be expanded and summed for all the pairs of variates
in the ith subsample, viz.,
3' (x  x)(y  y) = S» (x  x^Cy  f ± ) + n (%  x) (y ;l  y) .
It is noticed that the two middle terms of the expansion become zero for the summa
tion. The summation is taken again to include all the subsamples, viz.,
!fe'(x  x)(y  y) = S (x  x)(y  y) == Ss»(x  x i )(y  y ± )
1 1
+ n  [± ±  x)(y x  y)  .  (2)
The total covariance or correlation, S (x  X)(y  y), may be most easily computed by
this formula:
S (x  x)(y  y) = Sxy  (Sx)(Sy )  (5)
m N
means ,
The term, nS(x^  x)(y i  y) , which measures the covariance between subsample
can be computed as follows:
m m m a X
nS^  x)(yi  y) = nS x^i  (Sx)(Sy) = S X 2 y£  (Sx)(Sy) (k)
1 1 N 1 a ^ a n
where x a and y a are abbreviations for S'x and S»y the sums of the variates in a
single subsample.
m
The term, SS'(x  %)(y  fi), which measures the covariance within the subsamples,
can be found by subtraction.
The computation is analogous to the ordinary case of analysis of variance. In fact,
it should be incorporated with it for each variable separately. An illustrative
example will make the computation and analysis clear.
113
III. Computation of Co variance \j/
The data used to illustrate thin problem involve height measurements of 5 plants, from
each of 13 inbred lines of sweet clover together with a determination of the percent
age of leaves ("by weight) on each of these plants. The data are given in table 1 for
height in inches (x) and per cent leaves (y) for each of the 65 plants..
Table 1. Data on Height and Percent Leaves of 5 Plants from each of 13 Lines of Sweet
Clover
Plant
Numb
er
Toi
,el '
Line
1
2
3
1+
5
Height
Sx
Leaves
No.
X
7
x
y
X
X
y
y~
5
Sy
(In.)
do)
(In.)
(*)
(In.)
(i)
(In.)
(*)
(In.)
W
1
63
^
.66
38
39
ho
62
39
69
1+0
319
202
2
TO
33
77
37
6k
39
53
30
61
1+0
3?5
192.
3
51+
37
51
50
56
1+9
61
35
56
1+9
2 T 8
220
k
i+o
50
39
kk
lOfr
k2
38
1+3
1+5
1+5
206
22)+
5
30
k9
1+0
30
■UI4
k2
39
1+3
1+0
kk
199
23O.
6
1+1+
1+8
50
30
54
kk
5t
1+2
•36
kk
238
228
7
58
58
60
38
58
1+2
60
1+0
6k
ko
300
198
8
5*i
142
52
i+8
Jib
1+0
52
1+8
kb
kl
230
225
9
52
1+1
36
39
52
1+2
kb
1+3
1+8
1+2
231+
207
10
38
1*0
kS
1+1
52
39
52
1+0
60
39
270
199
11
63
kl
$h
1+2
38
37
r O
3d
1+0
5'!
ko
292
200
12
1+8
50
} + 5
33
1+3
c"0
1+1
k9
l+l
53
220
239
13
1+3
1+7
31
in
1+5
 f
;:>o
lll
1+3
l,k
i+6
20l+
233
Tot,
3375
2819
Since there was no replication of these lines the total variability will be divided
into only two components: (l) between lines end (2) between plants within lines.
Let the height of plants be designated as (x) and. the per cent leaves be designated
as (y) .
The sum of squares for total variation in height of plants will be:
S(x 2 )  (Sx) 2 /lT = 180,831.0  175,21+0.1+ = 3590.6
The sum of squares for variation between lines is calculated from the sums of five
plants per line as follows:
S(x 2 a )  (Sx) 2 = 89 8,1+67  175.21+D.1+ = 1+1+53.0
J— N 5
In like manner the total sum of squares for per cent leaves will be:
s(y 2 )  (Sy) 2 = 123,7V?  122,257 = li+89.1
N
The sum of squares for the 13 lines in per cent leaves will be:
S(y 2 .j)  (Sy) 2  613,715 • 122 , 2379  88I+.7
~5~~" N 5
■*■ This illustrative example is one prepared by Dr. F. R. Immer ;) with minor modifica
tions .
115
The sum of products for total variation will "be obtained "by multiplication of each
plant height by the per cent leaves on that plant. The results are then summed.
This will be:
S(xy)  (Sx)(Sy) = 11*, 697  1*6,371.2 = l67*.2
K
The sum of products for variation between lines is obtained by a similar process,
viz.,
S(x hyi )  (Sx)(Sy) = 724,01*  1*6,371.2 = *1568.*
5 N 5
The analysis of variance and covariance table can now be constructed as given in
table 2.
Table 2. Analysis of
variance and
co variance
Variation due to:
D.F.
Sum of Squares
x 2 xy
due to:
1*
Mean Sq. due to:
x 2 y 2
Lines (Between)
Within Line 3 (Error)
12
52
**53.0 1568.*
1137.6 105.8
88*. 7
60*.*
371. 08**73 .72**
21.88 11.62
Total
6*
5590.6 167*.2
1*89.1
**Exceeds the 1 per cent points.
The sums of squares and sum of products for variation between plants within lines is
obtained by subtraction.
Differences between lines with regard to height of plants (x), and percentage of
leaves (y), may now be tested separately for significance in the ordinary manner. It
is noted that these lines were significantly different in both height of plants (x)
and per cent leaves (y), the mean square for lines compared with error being greater
than the 1 per cent point.
However, there is no method to determine the significance of covariance itself (xy).
That is determined by tests of significance performed on correlation or regression
coefficients calculated from it. This problem will be considered next.
IV. Calculation of Correlation and Regression Coefficients
The coefficients of correlation can be calculated directly from the sums of squares,
since
r = S(x  x) (y  y)      (5)
Vs(x  x) 2 Vs(y  y) 2
By substitution of the sums of squares and products for variation between lines,
given in table 2, one obtains:
r= 1568.* = .790
JW&O 788*77
The other correlation coefficients can be calculated in like manner from table 2,
merely by substitution of the sums of squares and products found in the appropriate
row in the table, for the source of variability to be considered.
The coefficient cf regression of y on x will be given by b' S(x  xUy  v) i.e.,
prediction of y from x. S (x  \)' d
116
By substitution of the sum of products and sum of squares for "lines" from table 2,
b = 1566.U = .5522..
Wyy .
The correlation and regression coefficients are given in table 5.
Table 3* Coefficients of cor relat ion and regression
Correlation between Regression of
height (x) and per per cent leaves on
Variation due to: ^JE.' cent leaves (y) _ height (y on x )
Lines (Between) 11 .790** .3522
Withi n Lines _J>1  . XjJjf .0930
Total'"' "6s[_ "" ".580 ** ".299^ "
**Exceeds the 1 nor cent point of Fisher's table V. A. o z  1
VN  3"
From Fisher's table V.A. it is seen that r ~ .790 is greater than the expected value
of r for ? = ,01. The chances are, therefore, in excess of 99 ; 1 against the occur
rence of so large a correlation coefficient thru errors of random sampling from un
corrected material. The degrees of freedom for Fisher's table V.A. are 2 less than
the number of pairs in the sample and would, therefore, be one less than the degrees
of freedom in table 2 .
The correlation coefficient within Lines, r • .114 is not significant. The degrees
of freedom are 51 in this case.
V. Tests for Signif i oance for P egr eo s 1 on Coefficients
The regression coefficients can be tested for significance by means of an analysis of
variance or b? means of a "t" test. The former method will be illustrated first.
( a ) Test by Analysis of "variance
Suppose there exists a linear regression of percentage of leaves (y) on plant
height (x) . Then y 0; . the estimated percentage of leaves from a sample of N pairs of
values of y and x, is given by the regression equation:
y e = a + b(x  x). .•."".'    (6)
In this equation, a = y and b = S(y  y) (,X  x) are estimates of the true mean per
slxlc'F"
centage of leaves and the true regression coefficient, respectively.
Since the regression equation can be written as y  y e  b(.X  x), it is
apparent that :
S(y  y) 2 = sfy [y fc  b (x  *)] [ 2
= £(y  y e ) 2 + 2bS(y  y e )(s  x) 4 b 2 S(x  x) 2
Due to the fact that the middle term on the right is zero, v
V Consider S(yy e )(xS)
Since y Q = 4 y + b (x  x) , we have:
S [ y  y  b(x  x)  (x  x) or S(j  y)(x  x) bS(x  x) 2 .
It is clear that the whole expression is zero due to the fact that
b * S(y  y)(x j£j .
S (x  x) 2
S (y  y) 2 = S(y  y e ) 2 + b^Cx  x) 2
117
(7)
Thus, the sum of the squares of the deviations of the percentage of leaves has "been
analyzed into two components, one dependent on b, and therefore ascribable to regres
sion, and the other a sum of squares that represents deviation from regression or
residual.
Since b = S(y  y)(x  x) , it is obvious that the value of b S(x  3c) 2 , the component
S(x  £)' d
ascribable to regression will he:
b 2 S(x  x) 2 = Cs(x  Z)(y~ 7)32  (8)
S(x  if
This procedure is now applied to the illustrative problem. The values from
table 2 will be used to compute total regression:
S(y  y) 2 = 1489.10
[s(x  x)(y  y^l 2 = (1674. 2 ) 2 = 501.57
s(x  x) 2 5590 . 6
The analysis to test the significance of total regression follows in table 4.
Table 4. Analysis of Variance to Test the Significance of Total Regression
Variation
D.F.
Sum of Products
Mean Product
F value
Due to Regression
Deviations from Regress.
1
65
501 .37
987.73
501.37
15.68
31.98**
Total
64
3M9.10
The total sum of squares for y (leaf percentage) is taken directly from table 2.
Here y is used as the dependent variable, i.e., y (leaf percentage) is predicted from
x (plant height) which is known. The sum of squares due to deviations from regres
sion is obtained by subtraction, i.e., .U189.IO  501.37 = 987.73. There will be one
degree of freedom due to linear regression with a remainder of N2 degrees of freedom
for deviations from regression. It is also to be noted that N2 is the number of
degrees of freedom used to test the significance of r (Fisher, Table V.A). It is
obvious from table 4 that the regression coefficient is highly significant, since the
"F" value exceeds the one per cent point. The same conclusion was obtained when r
was tested for significance. In fact, the two tests for significance are equivalent.
When the correlation coefficient is significant, the regression coefficient must be
significant, and vice versa.
To test for the significance of regression between lines, the values already
computed for that source of variation in table 2 are used:
S(y  y) 2 = 884.7
CS(*  x)( y r J±P a L4168J4_}2 . 552.4
s(x  s)2 TT53 .0
The values are summarized in table c j\
Table % Analysis of Variance to Test the Significance of Regression Between Lines
Variation D.F.
Sums of Products
Mean Product
F value
Due to regression 1
Deviations from regress. 11
552.4
332.3
552.40
30.21
18.29**
Total 12
884.7
1.18
It is thus evident that the regression between lines is extremely significant. The
regression within lines will not ho tested for significance since r is not signifi
cant (See table 3) •
: '0 The "t" Test of. S ignificance
Regression coefficients may he tested by means of the "t" test also (See
Fisher _, 193 J v Pp. 126137) • As an illustration, the significance of the regression
of y on x between. lines may be tested. From table 3 ; b = r0'.yj>22 } S(x  x) =
44.53.0, and s(y  y)' c = 83h . 7 .
Then .,
where
Tlien,
t  b Vs (x  x) 2     ..  ~, r  (9)
P
s 2 = S(y  y) 2 j^^ix^ £1? ~ ~™  (10)
m  2
06^.7 zJ&iE&^MShQl  22LJL = 5° 21
132 11
.496
= °« 3^22 V4455 .O = 4,276 for 11 D.F.
J
r  .496
From the "t" table it is obvious that the observe! t value exceeds the 1.0 per cent
point .
Since fT = t for one degree of freedom, it is noted that /F = VlS.39 ■ =
4.277 (from table 5). Thus, it is apparent that tests of significance of regression
coefficients by means of the analysis of variance arid the "t" test are equivalent.
Moreover, they give the same result as tests of significance of the correlation
coefficient (Table V Ag Fisher, 193'+) •
VI. Substitution in Regression Equation
The regression equation is usually expressed as y e = y + h(x  x) .
For such a regression between lines one may substitute y  kj.yjjX = 51.92 and b =
.3p22. The mean values of y and x are obtained directly from table 1 "oj division of
the totals by 65. The value of b is taken from table 3. Numerically y Q  43.37 
0.3522 (x  51.92). This regression equation can be simplified to y e ~ 61.66  O.3522
x, where x is any value of plant height. In table 6 is' given the mean height of' each
line, the mean leaf percentage of each line and the leaf percentage predicted from
plant height by means of the equation above.
Table 6. Observed Mean Height', Mean Leaf Percentage and Predicted Leaf Percentage of
the 13 Lines of Sweet Clover.
Observed
Observed
Predict el
Observed
Observed
Pr
edict ed
Line
mean
mean $
mean $
Line
me an
mean $
mean \
ho .
height
leaves
leaves (y e )
No.
he i flit
leaves
le
aves (y e )
(x)
• (y)
U)
(y)
1
63.8
4o.4
392
8
50.0
45.0
44 .
2
65.O
33,4
58.8
9
50.8
. 41.2
43.8
3
35.6
44.0
42.1
• 10
s4 .
39.8
42 . 6
4
4l .2
44.8
47.1
11
58.4
40.0
4.1.1
5
393
46.0
47.6
12
44.0
31.8
46.2
6
7
51.6
60.0
45.6
39.6
43.5
40.5.
13
40.8
47.0
V7.3
119
The differences between observed mean leaf percentage and predicted leaf percentage
in table 6 represent errors in prediction. The sum of squares of these differences
would be given by S(y  y e ) where y represents the observed mean leaf percentage and
y s tho predicted value. This quantity can bo computed from table 5 hy subtraction of the
observed and predicted leaf percentage, these values being squared and added to give
S(y  y e )  66. ')k. "Now this sum of squares is based on means of 5 plants per line
while the analysis of variance in table k was on a single plant basis. Therefore,
multiplication of 66. ^k by 5 to place it on a single plant basis gives 332.7 • This
agrees with the sum of squares due to deviation from regression, i.e., 332.3 consider
ing that the predicted leaf percentages have been computed to only one place of deci
mals.
It may be noted also that s 2 used in the "t" test could be written s 2 = S(y  y ft ) 2 ,
m  2
since S(y  y e ) 2 = S(y  y)~  b w S(x  x) , the latter form being simpler for compu
tation purposes.
While the application of analysis of variance and covariance to correlation and re
gression problems has been illustrated here with data from a very simple experiment,
it is evident that it is equally applicable to problems of any degree of complexity.^
The analysis of variance and co variance are keyed out for the particular problem
under investigation after which the correlation and regression coefficients are cal
culated for scay component of the total variability. The tests of significance are
made in a manner similar to the ones illustrated.
VII. Use of Covariance
The analysis of covariance is often successfully applied in an artificial reduction
of experimental error in certain types of experiments where preliminary or uniformity
trial data are available. There may be factors which it is impossible to equalise
satisfactorily between the different treatments, and yet there may be reason to sup
pose that greater accuracy would arise from their equalization, were that possible.
Availability of preliminary data may provide the basis for such equalization.
The possible use of data from a previous uniformity trial to reduce errors due to
soil heterogeneity in the experimental years has been given considerable attention in
field trials in recent years. The assumption is that soil fertility is constant from
year to year. Thus, a significant correlation between the seme plots in successive
years may be used to reduce the error in the experimental year. The regression equa
tion is applied to predict the yields in the experimental year from the yields of the
same plots grown under uniform treatment in the previous year. The deviations from
the predicted yields should then contribute to the error of the experiment . Methods
to utilize information from previous crop records have been outlined by Fisher (193*0*
Sanders (1930), Eden (1931), and by Wishart and Sanders (1935). With annual crops,
Summerby (193*0 found that it was not worthwhile to sacrifice a year to a uniformity
trial in order to obtain information to reduce the error in the experimental year.
The method seems to have the greatest possibilities with perennial crops. (See
Fisher, 1937).
Another possible application of covariance arises where stand counts are available
in addition to yield. Mahoney and Baton (1939) have made such an application. Stand
counts may furnish a good index of plot variability provided they have been unaffect
ed by treatment. Correction for stand, which can be made from the regression relation.
VFor a consideration of curvilinear regression and its treatment by the analysis of
variance, the reader is referred to more advanced works on the subject.
ISO
provides an adjustment of the data to what they would "be if all plots had the name
number of plants (proportionality assumed). When, yield, is related to plant number it
is obvious that the experimental error will be decreased when this factor is taken' '
Into account and a correction made for it. It is first necessary to determine whothe?
or not such a relationship exists.
The simpler aspects of covarlance as applied to between and within groups have al
ready "been considered. The method will be used here for ordinary field experiments
where the total variation is sub divided into more than two parts. An illustrative
example used by Fisher (193'M will be followed.
VIII. Use of Preliminary Trial Data for Srror Reduction
_Si '
Some, data collected by Eden (19,31) Sf on tea will bo used to Illustrate the calcula
tions for cova.ria.iice for preliminary and experimental yields. Four "dummy" treat
ments for yields of tea expressed in per cent of the mean in a randomized blocl: ex
periment are given in table 7. '
Table 7. Preliminary and Experimental Yields of Tea Plants
Pr el iminary ( x )
or Blocks _ Treatment
Treatment Experimental (y) 1 2 5 4 Total Mean
m.
A x 91 118 109 102
y 6? 121 114 107
3 x 68 94 105 91
y 8l 93 106 92
C x 88 110 115 96
y 90 106 111 102
D x ■ 102 109 94 88
Q*5 ] "i 1
s
7 95 10! .9:
Block Total x 369 k : )l 423
. . •■ . y . 349 ' k$k 424
420
103.00
427
106.75
378
94 . 50
.572
93.00
409
102.25
409
.102.20
593
98.2p
'•SQp
98.00
loOO
1800
The preliminary yields will be designated as x and the experimental yields as y in
the subsequent calculations. ' '■ '■■''
(a) Analysis of Vari ance and Covarlance for Pr eliff d.narv and Experimental ^Yields .
The sums of squares for preliminary yields:
Total: Sx2  (g x)2  1326.0
" N
Treatments: Sx^  ( S: 0~  253.3
Blocks: Sx (Sx) 2 ~ 743.0
Sums of squares for experimental yields:
Total: Sy2  (Sy) 2 = 2040.0
,. ' IT ' •■.;."■■
■ Blocks: Sy2  (Sy)£ = 1099 5 '_
Treatments: Sy?  (Sy)2 = 414.5
: *F ' N
lfM
Cited by P. A. Fisher (1934)
121
Suns of Products:
Total: Sxy  (Sx)(Sy) = 1612.00
Treatments: Sx t y t _ (Sx)(Sy) = 32325
~ t »
Blocks: Sx t y t  (Sx)(Sy) = 837.
The above results are incorporated In table 8.
Table 8. Analysis of Variance and Covarianco
Variation
due to
Sums of
. Squares
D.F. U) (y)
Sums of
Products
(*y)
Mean
Square s
(x) (y)
F Value
(x) (y)
Blocks 3 7^5.0 1095.5 837.OO 2^8.33 365.17
Treatments 3 253.5 hlk.5 323.25 81+. 50 138.17
Error 9 5275 530.0 ^5175 58.61 58.89
k.2k* 6.20*
l.M 2.35
Total
15 1526.0 2C40.0 1612.00
From this analysis it is clear that no significance resulted between yields in case
of the "dummy" treatments, while a considerable degree of soil heterogeneity evidently
exists because the variation between blocks proved significant for both the prelimi
nary and experimental data.
It is now proposed to test the covariance as a basis to provide a correction
for the mean experimental yields in an effort to reduce the soil heterogeneity effect
further. The analysis of covariance is given in table 9*
Table 9 Analysis of Covariance and Test of Significance of Adjusted Experimental
Means
Variation due to
D.F.
Sum of
Squares
(x)
Sum of
Product
Sum of Errors
s Squares Sums
(y) Squares
of Estimate
Mean
D.F. Squares
Blocks
Treatments
Error
■7.
>
3
7^50
253.5
527.5
837.00
323.25
1*51.75
1095.5
klk.5
530.0 li+3.12
8
17.89
Total
Tr. + Error
15
12
1526.0
781.O
1612.00
775.00
20^0.0
9kk. 5 175.^5
11
15.95
Test of significanc
e for
adjusted ■
treatment
means 32 . 3?
3
10. 78 1
3F
» 17.89/H
D.78 a 1.
66 non significant
Since the total has been broken down into more than two parts, it is necessary to
form a new total which contains only the two effects under study, viz., treatment and
error. This new total is in the line, treatment + error, in table 9. The degrees of
freedom, sums of squares, and products are added to obtain the appropriate numbers.
The sums of squares for errors of estimate, S(y  y e ) 2 , are calculated by use of the
principle of subtraction, viz.,
122
[Sy 2  (Sy) 2 /uJ  fsxy  (Sx)(Sy) /n] 2 , in the lines for error, arid treatment
Sx 2  (Sx) 2 /K
+ error. These computations are as follows: •.".
(1) Error: 530.0  (k51.75) 2 /527.5 = 1^3.12
(2) Treatments + Error: 9I&.5  (775) 2 /f8l = 175. h O
The sums of squares for error is subtracted from that for treatment + error to yield
the sura of squares appropriate for the test of significance for the adjusted treat
ment means , viz., 175.45  143.12 = 32.33.
Ill this particular case, the mean square for adjusted treatment means is not signi
ficantly different from error since "dummy" treatments were used.
(h) Calculation of the Regression Coefficient
The regression coefficient (b) is calculated from the values in table 9« The
regression required is the regression of y on x in the row designated error. Since
the regression coefficient is the ratio of thus products to the sums of squares of the
independent variable ,
b = Sxy  (Sx)(Sy)/N * ^51 .75 = 0.8564
Sx2  (Sx) 2 /N 527.50
The significance of the error regression, b = 0.&)6h, should be tested at this point.
Unless it is significant, there will be little advantage to use it to reduce the
error for the experimental year. The sum of squares due to linear regression will
be:
(Sx)(Sy)/K 2  (kJL. 75) 2  386.8
Q£ 33
Sx 2  (Sx) 2 /W 527.50
The test for significance is summarized in table 10.
Table 10. Test of Significance for Error Regression
Variation Sum? Mean
due to Formulas .. D.F. Squares Square
TP
Regression [Sxy  (Sx)(Sy)/ft] 2 1 386.88 386.88 2 1.6 3
Sx2 ■ (Sx) 2 /lf rr
Deviations from r , v o't , . < no
regression Sy 2  J&dg YSSSLzJ^) (g.?lM 2 1^3.1? 1786
_L ' g J Sx 2  (Sx) 2 /N
Total for Error Sy 2  (Sy 2 /W 9 530.00 58.89
Error for adjusted yields.
The observed F value is highly significant. It indicates that it will be worth while
to proceed with the correction of the experimental test data on the basis of their
regression on the preliminary yields.
( c ) The Adjusted Treatment Me ans
The adjusted treatment means can be calculated and. compared with the unadjust
ed. The formula for the adjusted values is, y e  bx, where y is the individual treat 
' "meet in the experimental year. The computations are given In table 11. The mean
yields per treatment of the original data, x and y, are taken directly from table 8.
123
Table 11. Calculation of Mean Yields of Treatments in Experimental Test corrected for
Yields in Preliminary Test.
Mean Yield Deviations Mean Yield Corrected Yields for
Treat Preliminary from Mean Product 1 Experimental Experimental Test
ment Test (x) (x  x) b(x  x) Year (y) y e  "b(x  X)
A
105.00
5.00
I* .28
106.75
102. J+7
B •
9^.50
5.5O
U.71
93.00
97.71
C
102.25
2.25
1.95
102.25
100.32
B
93.25
1.75
1.50
98.OO
99.50
Gen.
Mean
100.00
0.00
0.00
100.00.
100.00
h = Sxy  (Sx)(Sy)/N = , 0.85&
Sx 2  (Sx) 2 /N
The regression equation for error for x on y is as fo'lows:
y e = y + b (x  x> = 100.0 + 0.856^ (x  100.00) = 0.Q r )6kx + 1^.36
The graphical representation is shown "below, the points for the determination of the
line determined "by substitution in the regression equation.
Let x = 92. 5, y e = 95.58. Let x = K>7.5,y Q = 106.^2.
110
Yields in
experimental
year (y)
90. 95 100 105 110
Yields in preliminary test (x)
(d) Standard Error of a Diff erenco
The standard error of a given difference between the corrected mean yields
is given by Wishart and Sanders (1936) as follows:
cr _
ex
= /2s 2
* 2 (x
xo
) 2
(11)
n
A'
where s 2 = the variance of the corrected yields (17.89), n =. the number of plots per
treatment (h) , A' = the sum of squares for error in the original preliminary trial
(527o) , and X]_ and %> = the means of the preliminary treatment plots being compared
(105.00  9^.50 = 10.50).
124
For treatment A and B, the mean difference of the corrected yields (table 11) is
102. '+7  97.71 = 4.76. The standard error is computed as follows;
°"c* = /2(1T.S9) + (17769) (10. 50) 2 = 356
■V * 527.5
^/^oX = 4.76/3.5o = 1.34, a nonsignificant value.
(e) Factors in Use of Independent Variable
The investigator usually wishes to know when it is worthwhile to introduce
the independent variable into the experiment. This question is answered by Snedecor
(1937) who states that three items will aid him. First, the list of actual and ad
justed means. Sometimes the rank order of adjusted means is quite different from
that of the unadjusted and the shifts may be interpreted. Second , a comparison of
the sum of squares of errors of estimate (table 8) used to test treatment signifi
cance, 32.33, with Sy2  (Sy)2/w = klk .5. The latter is far greater' than the former.
Third, the change in precision of the experiment due to the adjustment of the error
sums of squares. This is indicated in table 8. The sum of squares, Sy2  (Sy) £ /w =
530.00 with 9 degrees of freedom, is analyzed into two parts, one with a single de
gree of freedom that measures the variation attributable to regression, the other 8
degrees of freedom being assigned to error. The mean square for error is reduced
from 53.59 to 17.89, which is highly significant. These factors will enable the in
vestigator to decide whether to retain the independent variable in similar experiments.
It has already been mentioned that the use of preliminary uniformity data to reduce
the error in the subsequent experimental test may be useful in perennial crops, but
probably is not worth while for annual crops.
"References
1. Bartlett, M. S. A Note on the Analysis of Covariance. Jour, Agr. Sci.
26:438491. 1936.
2. Cox, G. Mc, and Snedecor, G. W. Covariance used to Analyze the Relation between
Corn Yields and Acreage. Jour. FarmEcon., l8:597'607? 193^.
3. Eden, T. Studies in the Yield of Tea. I: The Experimental Errors of Field
Experiments with Tea. Jour. Agr. Sci., 21:547573. I.93.I.
4. Fisher, R. A. Statistical Methods for Research. Workers . Oliver and Boyd, 5th
edition, pp. 257272. 1934.
5. The Design of Experiments, Oliver and Boyd. 2nd Ed., pp. I72I89
1937 .
6. Garner, F. II . Grantham, J., and Sanders, E. G. The Value of Covariance in
Analyzing Field Experimental Data. Jour. Agr. Sci., 24:250259. 1934.
7. Goulden, C. H, Methods of Statistical Analysis. Burgess Publ . Co., pp. 151157,
and 185196. 1937.
8. Immer, F. F. A Study of Sampling Technic with Sugar Beets. Jour. Agr, Res.,
44:663647. 1932.
9. , and Raleigh, S. M. Further Studies of Size and Shape of Plot in
Relation to Field Experiments with Sugar Beets. Jour. Agr. Res,, 47:591598.
1933 .
10. Mahoney, C. H., and Baten, V. D, The Use of the Analysis of Covariance and its
Limitation in the adjustment of Yields Based upon Stand Irregularities. Jour.
Agr. Res., 58:317328. 1939
11. Sanders, H. G. A Note on the Value of Uniformity Trials for Subsequent Experi
ments. Jour. Agr. Sci., 20:6373. 1930.
12. Snedecor, G. W, Statistical Methods. Collegiate Press, pp. 219241. 1937.
125
13. Sumtaerby, P. The Value of Preliminary Uniformity Trials in Increasing the
Precision of Field Experiments. McDonald Col. Tech. Bui. 15 . 193^.
ll+. Wishart, J. and Sanders, E. G. Principles and Practices of Field Experimentation.
Emp. Cotton Growing Corp., pp. 1+556. 1935.
Questions for Discussion
1. What is covariance? Where useful in experimental work?
2. Interpret the use of the analysis of variance for the determination of signifi
cance of the regression coefficient.
3« How is the error for linear regression computed?
h. How can covariance he used on preliminary trial data to reduce the error in the
experimental year?
5. Discuss conditions in field experimentation where it might he useful to use
preliminary trial data to reduce the error in the subsequent test.
6. What assumption is made in the correction of stand "by covariance? What precau
tions are necessary?
7. Upon what is the error of estimate based? Explain.
8. What does it mean when the difference to adjust treatment means actually is less
than the mean square for the error of estimate for error?
9. Name 3 types of agronomic tests where covariance might prove useful. Give the
reason in each case.
Problems
1. The yields of soybeans in a randomized block experiment with split plots are given
below. Let x represent the yield of hay in tons por acre and y represent the
yield of seed in bushels per acre.
The total yields of the k plots of each spacing are assembled below.
Bu. of seed per acre (y) Tons o f h ay per ac re (x)
Width
Spa
cing w:
Lthin rows
Width
of rows
Sp f
acing within rows
of rows
1/2"
1"
2"
■ r
Sum
1/2"
1"
3"
Sum
16"
89.3
91.8
79.6
■ 88.6
3^9.8
16"
11.1+0
10.72
9.63
9.68
1+1.1+3
20"
92.7
85.6
37.2
87.I
352.6
20"
II.3I
10.06
9.73
9.31
1+0.1+1
2k"
90.6
82.3
8H.3
80.7
3379
2k n
10.02
9.21
9.00
8. to
36. 61+
28"
86.0
83.O
82.1+
78.3
329.7
28"
0.62
9. in
°.09
8.28
36.1+3
32"
85.I
78.1+
7^.6
72.9
311.0
32"
9.53
8.72
8.1+5
777
3^.52
1+0"
78A
70.7
71.7
69.2
290.O
1+0"
8.31
8.19
73^
759
31.93
Sum 522.6 1+91.8 1+79.8 1+76.8 1971.O Sum 60.7!+ 56.3I+ 53.2^ 51,0> 221.36
The analysis of variance for x and y are given below
Variation due Correlation Regression
to: D.F. ( yj) (xx)(yy) (xx) of x and y of y on x
Blocks 3 IO.I+370 .II+38
Width of rows 5 182.0500 3.968!+
Error (a) 15 52.55I+2 .J___3
_______ 23 22 5. 01+12 1+. I+655
Spacings 3 5^.7512 2.2108
Width x spacing 15 30.1+300 .2389
Error (b) ___+ 268.6038 1.598^
Total 95 578.8262 ' 8.5115
126
(a) Calculate an analysis of co variance.
(t>) Calculate the correlation coefficient for tho different lines in the
'analysis of variance and covariance.
(c) Do the same for the regression coefficient of y on x.
(d) Test the significance of the correlation coefficients 'by means of Fisher's
table V.A. Mark the coefficients which exceed the c ) per cent point with
one asterisk(*) and those which exceed the 1 per cent point with two
astericks (**) .
(e) Test the significance of the regression for error ("b) "by means of an analysis
of variance. Get the fl? also.
(f) Test the significance of the regression for error ("b) "by means of "t" test.
(g) Test the significance of the correlation coefficient for error (b) by means
of the "t" test ; given by Fisher in section jk of his book.
(h) Calculate the mean yield of seed In bushels for the six different width of
rows. Calculate the predicted mean yields for each width, using the regres
sion for width of rows.
Some data on number of sugar beet plants per plot and yield in tons per acre are
given by Snedecor (1937) for a fertilizer experiment conducted in a randomised
block test. The data for p replications are given below:
Fertilizer No. (x) or
Applied Yield (y)
None
x
y
Block
1 2 5
I83 176 291
2.M 2.2p k.^Q
Treatment
Sum
Sums Sums
S c: uai'O s Product s
x
7
356 300 301
6.71 3. Mi ] +.92
K
x.
y
22 k
p.<£
2p8 2kk
k.lk 2.32
?K
x
y
6 . 34
303
.22
. 1,
PIT
x
y
371 33^ 332
6.kB 7.11 5.88
KN
NPK
Block Sums
x
y
X
y
X'
y
230 221
3.70 3.2*f
237
2.82
322 367 k0(
6.10 7^8 7.37
Svan. Squares
Sums Products
x
y
Due to the fact that the number of plants varies it is necessary to examine the
effect of the variable stand and to estimate the yields on the basis of equal numbers
of plants. Calculate as follows:
127
(a) Yield in a simple randomized "block experiment.
(t>) The analysis of covariance of stand and yield.
(c) Calculate the test for significance.
(d) Give the conclusions for the test.
3. A sugar "beet variety test was conducted at Rocky Ford as a randomized block exper
iment in which tho number of plants differed in each plot. The yields were taken
on the basis of competitive plants per plot, and also on the basis of all the
beets in the plot . The object of the experiment was to determine the yields of
the different varieties. Since the number of beets varies from plot to plot, it
is desired to examine the effect of the variable stand and to estimate the yields
on the basis of equal numbers of beets. The data follow (unpublished data from
0. W. Deming) :
Variety
No
Yi
.00
eld
or
(y)
Block
Treatment
No.
1
3
4
5
6
Totals
1
X
y
243
17.49
217
I8.65
227
1.4.39
210
16 .33
218
11.28
215
15.75
2
X
y
245
17.99
217
19.22
239
18.21
210
14.11
205
16.84
219
13.76
3
X
y
238
14.13
228
16.62
205
15.99
191
13.81
224
14. 80
211
13.00
4
X
y
254
20.19
223
21.45
I89
14.01
180
12.54
23.6
14.20
209
17.65
5
X
y
249
20.08
221
17.04
226
14.04
242
16.05
246
13.86
216
9.75
6
X
y
225
17.49
212
19.63
194
17.55
211
16.02
202
15.46
215
14.05
Block
X
Totals
y
Calculate the analysis of covariance and adjust the yields to a uniform stand
basis.
FIELD PLOT TECHNIQUE
PAST III
Field and Other Agronomic Experiments
CHAPTER XII
SOIL HETEROGENEITY AND ITS MEASUREMENT
I. Universality of Soil Heterogeneity
One of the difficulties in yield teste is the fact that uniform soil conditions
rarely exist, even over a small portion of any field. Soil variability has "been
noted "by many investigators, "but it was J. Arthur Harris (1915)(1920) who first pre
sented data to show its extreme importance in field experimentation. Lyon (19H)
states that it is "quite likely that productivity of plots change from year to year
even with the same treatment", altho the work of Harris and Schofield (1920) (1928)
and of Garber, et al. (1926)(1930) indicates a tendency for the differences in plot
yields to "be permanent .
A soil with differences so slight as to escape the most oh servant eye may have very
great effects on' plants which grow in it. Parker (1931) is authority for the state
ment that two plots of the same crop variety grown in "an apparently uniform soil
and treated alike in every respect may differ from one another in yield "by 20 per
cent or more solely as a result of differences in soil conditions." Small plots have
generally replaced large ones to correct for this condition, because it is obvious
that two plants of the same variety grown one yard apart are more likely to yield
alike than when 200 feet apart as probably would be true of oneacre plots. It is
impossible to avoid variation even under such conditions. Davenport and Frazor
(I896) report results with 77 variotios of wheat grown on plots two rods square.
Nine check plots of the same variety were systematically distributed over the area.
The variation in the check plots was so great that only 8 varieties yielded more than
the highest check, and but 3 lower than the lowest check.
Soils vary in texture, depth, drainage, moisture, and available plant nutrients from
yard to yard. After the analyses of large amounts of data from all over the world,
Harris (1920) concluded that soil heterogeneity was practically universal. He esti
mated it to be the most potent cause of variation in plot yields and the chief diffi
culty in their interpretation. In 1915 he. stated: "It is obviously idle to conclude
from a given experiment that variety 'A 1 yields higher than variety 'B 1 , or that
fertilizer 'X' is more effective than fertilizer 'Y 1 , unless the differences found
are greater than those which might be expected from differences in the productive
capacity of the plots of soils upon which they are grown." Even earlier than this,
Piper and Stevenson (1910) remarked that soil variability was so great that "doubt
was cast on the greater portion of published field experiments where yield was pri
marily involved." The yield differences must be large enough to overshadow soil
variation, or the experiment designed so as to remove its effect.
Much of the improvement in experimental methods for field experiments in recent years
has been brought about thru special devices to measure much of the soil fertility
variation and essentially eliminate it from the actual comparisons being made.
II. Uniformity Trial Data
Uniformity trial data ha.ve been used for the measurement of soil heterogenity as
well as for many other purposes in field experimentation. The usual procedure is to
plant a bulk crop, the area being later partitioned into small plots, usually of the
same dimensions. The same cultural operations are carried out over the entire area.
The yield of each plot is recorded separately at harvest. The usefulness of the
uniformity trial lies in the fact that the small units can be combined into larger
plots of various sizes and shapes in order to study variability. The variation in
132
yield over the field is due to soil heterogeneity, as well as to plant variation,
errors Inveighing, etc., (generally summed up as experimental, error). The; most
obvious use for the data is to provide information on the optimum size and shape of
plot. Uniformity trial data can also he used to compare the relative efficiencies
of different experimental designs, particularly in relation to a certain crop. Data
from previous uniformity trials may also he use! to reduce the error of subsequent
experiments laid down on the same plots.
The method offers promise for perennial crops where the same plants are concerned,
but offers little or no advantage for annual crops. A catalogue of uniformity trial
data has been published by Cochran (1937) «
Some agronomists conduct socalled blank trials (planted to a bulk crop) to observe
soil heterogeneity as a preliminary step in experimentation on a new field. Love
(1928) advocates such trials, especially as a preliminary to longtime experiments,
Th'ey afford an opportunity for the investigator to detect good and poor spots on a
field so that unsatisfactory areas may be eliminated. One objection to the blank
trial used in this manner is that it takes time. Time may be an Important element .in
an experiment .
Ill . Criteria for the Measu rement of Soil Tari abi I icy
Some accurate measure of soil heterogeneity may be desirable preliminary to seeps for
its c c rre ct i on .
(a) Correlation Coef f . ; cient
Harris (1915) supplied the first quantitative measure baseu on correlation.
his heterogeneity coefficient being an intraclass correlation coefficient, "dor use
of the formula,, the field must be planted uniformly to the sains crop and harvested in
small units. Harris grouped nearby plots. The number in a. group was arbitrary, it
being common to use 2 by 1, 2 by 2, and 2 by 3 fold groupings. The size of the
heterogeneity coefficient is influenced by the size of group. The more p] ots that
are put together, the greater is the correlation coefficient. The heterogeneity
coefficient is expressed on a relative scale from 0.0  1.0 so that comparisons from
field to field can be made directly. This coefficient measures the degree to which
nearby plots are similar in productivity. Should the correlation be sensibly zero,
the Irregularities of the field are not so great as to influence in the same direc
tion the yields of nearby small plots. The higher the correlation, the greater the
soil heterogeneity. One may grasp the significance when he remembers that the corre
lation coefficient multiplied by 100 gives the most probable percentage deviation of
the yield of an associated plot when the deviation of one plot of the group from the
general average is known. Hayes and Garber (IQ27), in explanation, state that in
"patchy" fields certain contiguous units tend to yield high while others sh. w a ten
dency in the opposite direction. Under these conditions a high correlation coeffi
cient results. "Where variability is due only to random sampling the correspondence
between contiguous plots will be counter balanced, bv lack of correspondence in others.
The same result can be obtained with the ordinary inter class correlation coefficient
as with the heterogeneity coefficient when a 2 by 1fold, arrangement is used,.
The analysis of variance can be used, to obtain the same result as with the hetero
geneity coefficient as ind.ica.ted by Fisher (195*0 . Intraclass correlation merely
measures the relative Importance of two groups of factors that cause variation. In
the calculation it is necessary to obtain these equalities:
"1? (x  x)2 = ( m  l) n S 2   ... (l)
nS (xb  x)* 2 = (m  1)b2 I 1 + (n • 1) r 1 •• (2)
1
L
133
vhere m = the number of arbitrary blocks and n = the
number of ultimate units within a block
The principal value of the correlation coefficient, either interclass or intra 
class, is to demonstrate that the fertilities of adjacent areas are correlated and
that variability exists in the field.
(b) Fertility Diagram
The suitability of a particular layout adopted in an experiment can be
judged to a considerable extent by a fertility diagram constructed from the individ
ual plot yields. This is possible from uniformity trial data. An example taken
from Crowther and Bartlett (1938) is given in Figure 1.
Figure 1
Variation in natural fertility at Bahtim, 193 ,+ (yield
in kantars per f eddan) .
w
IV . Computation of Heter ogeneity by the Analysis of Variance
Suppose a field J a divided into K email plots* all sown to the same variety. Some
uniformity trial data from Mercer and Fall (1911) on the grain yields of one acre of
wheat when harvested in 1/500acre plots will be used to illustrate the method of
computation. The yields in pounds per plot for the 24 plots in the northwest corner
are as follows :
3.63
ffl]
4. 15 i
1
1
4.06
Hl>,
5.13
4. 07
4.21 :
4.13
4.64
4.51
ffig
4.29 I
4.4o
HU,
4 . 69
3.90
4. 64 i
4.05
4.04
3.63
m 3
4.27 :
1
1
4 , 92
m,<
4.64
3.1b
3.55 ;
4.08
4.73
It is noted that the area is divided into an arbitrary number of blocks all equal in
size, i.e., there are 6 blocks with 4 ultimate plots in each.
Let x = the value of an ultimate plot unit.
I\T = the total number of plots = 24.
S(x.) = sum of all the ultimate plots  101.34.
X  mean yield of the ultimate plots  4.2308.
3(x 2 ) = sum of squares of yields of the ultimate plots = 434. 5582.
Then, the sums of squares are computed as follows:
Total = S(x 2 )  (J3x)f : = 434.3382  422.5933 = 4.9394
I
Between blocks = S( x £) (Sx) 2 ± 1?'2T. 9410  429.5938  2.3864
— : g; jj
Within blocks = 4.9594  2.3864 ^ 2.5730
The analysis of variance is as follows:
Variation D.F. Sums Squares
Between blocks 5 2.3864 = (ml)s 2 [ ]. j (n  1) rj
Within blocks 13 2.5730  (ml)s 2 (n  l) (1  r)
Total 25 4.9594 = (ml)ns 2
Wow, let m = the number of blocks, m1 = the degrees of freedom for blocks, n  the
number of plots per block, and. s' c  = the estimated variance.
Then m = 6, m  I = 5> and n = 4. . '
Since (ml)ns 2 = 4.9594, 20s 2 = 4.9594, and s 2 = 0.2480
From the formula for the sum of squares between blocks,
155
(m  l)s 2 [l + (n. 1) r] * 2.386^
or (5)(0.2480)(1 + 3r) = 2.386^
Then r = O.3082
V. Amount of Soil Heterogeneit y >
In his studios of soil heterogeneity, Harris (1915) (1920) used fields planted to the
same crppj "but harvested in separate small plot units. The relative productivity of
contiguous plots was determined.
(a) Variations in Yield in Same Sea son
Some of the results obtained by Harris are given by Hayes and Garber (1927):
Plot Size Investigator r
Crop
Characters
Wheat
Grain Yield
N Content
Oats
Grain Yield
Mangels
Boots
Loaves
Potatoes .
Tuber Yield
Corn
Grain Yield
55 by 5.5 ft. Montgomery O.605 ± 0.029
0.115 ± O.oMf
1/30 acre Eiesselbach OA95 ± O.O55
1/200 acre Mercer & Hall 0.3^6 ± 0.037
O.U66 ± 0.0U3
12 foot row Lyon O.3H f 0.0^3
0.03;^ acre Smith O.83O i 0.019
The amount of soil heterogeneity in rodrow trials was measured by Hayes (1925) at
the Minnesota Station in connection with a variety test. Four systematically distri
buted plots were used. To obtain the heterogeneity coefficient the average yield of
each strain in the trial was considered as 100. The yielding ability of each plot
was obtained by dividing its actual yield by the average yield of all four replicates
and expressing the result in percentage. By the ordinary method, correlations in
yielding ability of adjacent plots or of plots at any distance apart were determined,
The results were as follows for oats ; , spring wheat, and winter wheat:
Correlation Coefficients (r)
Factors Correlated Pi 1 ^ Spring Wheat Winter Wheat
Adjacent plots 0.572 ± 0.025 0.6l8 ± 0.023 0.552 ± 0.063
Separated by one plot 0A90  0.029 0.518 ± 0.028 O.293 ± 0.028
Separated by four plots 0.26^ ± O.oUl .kk9  0.03^ O.llU * 0.118.
Separated by ten plots 0.275  0.057 0A29 ± 0.060  
The correlation coefficient explains very little unless one knows the factors in
volved. However, it affords the best means to consider the amount of replication
that should be practiced.
Similar results were obtained by Garber, Hoover, and Mcllvaine (1926) in West Vir
ginia experiments. They found a marked correlation between the yields of oat hay in
contiguous plots. The correlation for the yields of replicated plots was sensibly
zero .
(b) Permanence of Differences
It is important to know whether or not there is a tendency for plots that
produce low yields one season to produce low yields the next season, etc. The re
sults of Harris and Scofield (1920) indicate a tendency for plots to yield in a
similar manner from year to year, altho there are some exceptions. Their data for
interannual correlations for hop yields are as follows:
13 6
series
1909
1910
1911
1912
1st and 2nd
Years
1st and 3rd 1st and ^th 1st and 5th 1st and. 6th
Years Years Years Years
0.580
O.768 i 0.051 0.662  0.07
0.577 ± o.o32 0.447 t 0.099 . 0.451  0.098 0.27^
0.3.15 * 0.111 0.126 $ 0.121
0.105 0.259 *0..115
O.IIJ4
0.061  0.12*;
0.062 x 0.123
0.511 i 0.111 0.703  0.062
0.597 ± 0.079
the result:
5 year study on
In a later paper, Harris and Schofield (.1928) gt
a uniform cropping experiment at Hunt ley, Montana. In general, a positive correla
tion 'between the yields of a series of plots was found thruout a period of years.
The plots which show a heavier yield one year will in general show heavier yields In
other years during the perior under investigation. Under some conditions negative
correlations were found which were interpreted as indicating the importance of a
preceding crop in determining the characteristics' of an experimental field.
Garber, et al, (1926) found some tendency for plots which produced relatively high
yields of oat hay in IO23 to produce relatively high yields of wheat grain in 192''.
The correlation coefficient for th ; two
was 0.564 * 0.056. The study was con
tinued by Garter and Hoover (1930) to determine whether or net the natural variation
in noil productivity among plots as revealed "by a crop uniformity test persisted af
ter an experiment is started that involves different crops and different soil treat
ments. They correlated the relative yields from duplicate oat plots in 1923 and the
relative average yields from, the same duplicate plots of other crops in a rotation
experiment from 1924 to 1029 (incl,). The data were as follows:
'r ;i value between
elds
.n
1 Op".
> and
1924
1925
1926
1927
1928
"i 000
130
150
1.26
128
120
15o
O.38 i 0.05
0.35 * 0.05
0.48 * 0.05
0.41 * 0.05
0.42 ± 0.05
C.27 t 0,03
age 1923 to 1929 (incl.)
60 ±
'hese correlation coefficients were all statistically significant, and
the d3.nerences m naturax
this ce.se) even though the
:.ncLieaec
that
proauctxvrcy may persist ov^r a
soil be subjected to different treatments
riod of years (five,
T. Cant
of
He^eroffei
Many fact or s may contribute to soil variation. Yields
P. i*1 1 1 "•" !~YT\ *"\ ' l"T»W TvH *~'f
raoii
has demonstrated by the use of the correlation cc
geneity is sufficient to influence experimental r
moiseuro, ana s<
If let erf
:ults.
crops from foot to ; ma;;
>il fertility.
■ris (1020)
that
nibstratum netero
{&) Soil Topog;
?he topography oJ
may direc
Indirectly influence the variation
in soil productivity. Steep hillsides are unadapted to e
rains gully the field and carry the fertilisers from plot
is apt to pond on certain area.s and Influence crop yields
troduced l>:j variation in the subsoil. For example, there are gravel pockets in the
subsoil on the Judith Basin (Montana) field station.
perimentation because heavy
to plot . Moreover, water
Sometimes errors are in
137
(b) Soil Moistur e
The water content of soil was studied by Harris (1915) on the U.S.D.A. Exper
imental Farm at San Antonio (Texas) . He took "borings 6 feet deep at 20foot inter
vals on a field 150 "by 26h feet in size. The coefficients ranged from r = +0.32 to
0.70, being statistically significant for each foot section of the upper 6 feet of
soil.
(c) Fertility Elements
The carbon and nitrogen content of soils was studied by Harris (1915) at
Davis, California. The heterogeneity coefficient for carbon was O.U17 * O.063, while
that for nitrogen was 0.^9^  0.057 • On blow sands at Oakley, the rvalue for carbon
was O.3I7 ± 0.068, and that for nitrogen was O.230 ± 0.072. Wide fluctuations in
nitrate nitrogen were reported by Blaney and Smith (1931) on l/30 acre plots. They
found that the probable error was usually greater than 5 P e ^ cent where less than 20
soil cores were considered. In fact, they recommended 50 soil samples on a 1/30
acre plot to reduce the error to approximately 5 V 0T cent of the mean. When soils
outside of Rhode Island were considered, they found that 6 to 8l samples were neces
sary to obtain a probable error that low. Some Colorado Station data show extremely
wide fluctuations in p. p.m. nitrate nitrogen on an irrigated soil. A 13 by 10foot
plot was sampled in 5 places to a depth of 6 feet. The nitrate nitrogen varied from
5 to 35 p. p.m. on this small area. It is obvious that variations in nitjrate nitrogen
can cause yield differences from area to area.
VII. Corrections for Soil Varia bility
Once soil heterogeneity is recognized, some means must be obtained to avoid or cor
rect its influence in field experiments. A decrease in size of plots and an increase
in the number of replications (as will be shown later) has been the general practice
to overcome soil variation. The repetition of .plots of varieties or treatments to
be tested against each other are scatter*! out so that they may sample the different
conditions of the trial area. One variety, for . instance, may be grown partly on_
favorable portions and partly on less favorable portions. This usually means that
the variety encounters somewhere near everage soil conditions. Efficient experimen
tal designs provide for the removal of a portion of variability due to soil. Arti
ficially constructed field plots were studied by Garber and Pierre (1933) over a 3
year period. They found that soil heterogeneity was largely removed. by a thorough
mixture of soil placed in 30 artificial bins. These soil bins were 9 feet k inches
by k feet 8 inches (inside area) by 2k inches, in height, and ye^e 0.001acro in area.
They obtained a probable error of a single determination in per cent of the mean of
3A for soybean hay, and 6.2 for wheat. They found, however, that the variation in
crops was still too high to make replication unnecessary.
VIII. Relation to the Experimental Field
Many early experimental fields were poorly selected because of the belief that an
experimental farm should contain many different soil types, i.e., the soil should be
extremely heterogenous. The Ohio Experiment Station was allowed to relocate after
the first 10 years due to the poor choice of the original site. (Thorne, 1909) For
all ordinary field experiments the land should be as uniform as possible in regard to
topography, fertility, subsoil, and previous soil management. However, extreme uni
formity may defeat the purpose of the investigator unless such soil is representative
of the area for which the results are to apply.
(a) Topography
A perfectly level piece of land is as undesirable for field experiments as
one with surface inequalities because water may pond on it. A slope of 1 or 2 per
cent will permit water from heavy rains to flow off uniformly and completely. A
138
slight slope is highly desirable on land to be irrigated. Some experimenters use
low land or "draws" and irregular areas for bulk crops or for seed increase plots.
(b) Previou s Soil Treatment
It is desirable to have soils which have had uniform previous treatment be
cause there may be a carryover effect of previous treatments. According to the '
American Society of Agronomy standards (1933): "When a field or series of plots has
been occupied by varietal or cultural tests of such a nature as to seriously increase
soil variability, one or more uniform croppings should intervene (or follow) before
it is again used for such tests. It is frequently helpful to arrange the plots at
right angles to the direction of the previous plots."
(c) Subsoil Conditi ons
When it is necessary to drain lands in the humid regions, the tile lines
should be located so as to influence all plots alike. They should run across the
plots rather than with them. In the case of soil fertility experiments, it is recom
mended that a soil profile be taken to a depth of 3 feet for each series of plots.
Before soil treatment experiments are begun, representative samples of the soil and
subsoil should be carefully taken for such analyses as may be desired for future
reference.
References
1. Blaney, J. E., and Smith, J. B. Sampling Market Garden Soil3 for Nitrates. Soil
Sci., 31:281290. 1931.
2. Cochran, W. G. A Catalogue of Uniformity Trial Data. Suppl . Jour. Roy. Stat.
Soc, 4:233233. I937V
3. Crowthcr, F.,, and Bartlett, M. S, Experimental and Statistical Technique of
Some Complex Cotton Experiments. Erap. Jour. Exp. Agr. 6:3368. 1938.
4. Davenport, E., and Frazer, V. J. Experiments with Wheat, I&88I895. 111. Agr.
Exp. Sta. Bui. 4l, pp. 153153. 1896.
3. Fisher, E. A. Statistical Methods for Research Workers. Oliver and Boyd.
5th Ed. pp. 210214. 1934.
6. Garber, R. J., et al. A Study of Soil Heterogeneity in Experiment Plots. Jour.
Agr. Res., 33:2552.68. 1926.
7. Garber , R. J., and Hoover, M, M. Persistence of Soil Differences with Respect
to Productivity. Jour. Am. Soc. Agron., 22:883390. 1930.
8. and Pierre, W. II. Variation of Yields Obtained in Small Artifi
cially Constructed Field Plots . Jour. Am. Soc. Agron., 2^:98105. 1953*
9. Harris, J. Arthur. On a Criterion of Substratum Homogeneity (or Heterogeneity)
in Field Experiments. Am, Nat., 49:450454. 1915 .
10. . Practical Universality of Field Heterogeneity as a Factor
Influencing Plot Yields. Jour. Agr. Res., 19:279314. 1920.
11. , and Schof ield, C. S. Permanence of Differences in the Plots
of an Experimental Field. Jour. Agr. Res., 20:333356. 1920.
12 . , , , . Further Studies 01a the Permanence of
Differences in the Plots of an Experimental. Field. Join. Agr; Res., 56:1541.
1928.
13. Hayes, K. K., and Garber, R. J. Breeding Crop Plants. McGrawHill, Pp. 5669.
1927 .
14. . Control of Soil Heterogeneity and Use of the Probable Error Concept
in Plant Breeding Studies. Minn. Agr. Exp, Sta. Tech. Bui. 30. 1925
15, Lovo, H. H. Planning the Plat Experiment. Jour. Am, Soc. Agron., 20:426432.
1928.
16. Lyon, T. L. Some Experiments to Estimate Errors in Field Plot Tests. Proc. Am.
Soc. Agron., 3: 89114. 1911 .
139
17. Mercer, W. B., and Hall, A. D. Experimental Error of Field Trials. Jour. Agr.
Sci., k: 107127. 19H.
18. Parker, W. H. Methods Employed in Variety Trials "by the National Institute of
Agricultural Botany. Jour. Natl. Inst. Agr. Bot., 3:522. 1931.
19. Piper, C. "V., and Stevenson, W. H. Standardization of Field Experimental Methods
in Agronomy. Proc. Am. Soc. Agr on., 2:7076. 1910.
20. Standards for the Conduct and Interpretation of Field and Lysimeter Experiments.
Jour. Am. Soc. Agron., 25:803828. 1933.
21. Thorne, C. E. Essentials of Successful Field Experimentation. Ohio Agr. Exp.
Sta. Cir. 96. 1909 .
22. Wishart, J., and Sanders, H. G. Principles and Practice of Field Experimentation,
Emp. Cotton Growing Corp., pp. 78, and 606y. 1935.
Que stion s for Discussi on
1. Discuss how soil heterogeneity might influence yield trials.
2. Why and how may small plots overcome the influence of soil variation?
3. What is a uniformity trial? How conducted?
k. What uses can be made of uniformity trial data?
5. How can correlation "be used to measure soil heterogeneity?
6. Fundamentally, what is the socalled "heterogeneity coefficient" used "by J.
Arthur Harris? How interpreted?
7. What evidence did Harris have that soil heterogeneity was universal?
8. What general results were obtained at the Minnesota Station when the yields of
adjacent plots were correlated? Those separated "by other plots?
9. Are differences in the productivity of plots constant from year to year?
Explain.
10. How may soil topography, moisture, and nitrogen account for soil heterogeneity?
11. What corrections can he used for soil variability?
12. Would artificial soil bins do away with the need for replication? Explain.
13. What precautions should be taken in the selection of an experimental field?
1*. Is extremely uniform soil always desirable for experimental work? Explain.
15. What is the value of a bulk crop preceding an experiment?
16. To what use would you put uneven and low land in an experimental field? Why?
Probl ems
One acre was planted uniformly to the same variety of wheat and harvested in units
l/500acre in size. (Data from Mercer and Hall). The 16 plots in the southwest
corner of the acre gave yields in pounds as follows :
3.87
*1.2I
3.68
*K 06
3.76
3.69
5.8*
3.6?
3.91
k.y
h,2l
V.19
3.5^
359
3.76
3.30
(a) Calculate the correlation coefficient by the analysis of variance for a 1 by
2 fold arrangement.
(b) Calculate the simple correlation coefficient for the same paired values.
2. Some unpublished data from the Akron Field Station give the average yields of corn
and oats (combined) in bushels for a particular piece of land for 20 years as
follows:
.uo
North Totals
113.9
398
ko.6
39.4
165.7
39.0
kO.k
377
35.0
152.1
32.9
kO . 5
395
36.7
149.6
yd. 6
1+1.3
41.1
35.5
156.5
35. 4
'13 . 1
37.3
30.5
146.3
West
Totals I89.8 205.1 196.2 177.1
(a) Calculate the correlation coefficient by the analysis of variance to determine
the heterogeneity from north to south, i.e., for a 1 "by k combination.
(b) What is the correlation coefficient for a west to east direction? Calculate
r for 1 by 5 combinations.
.(c) In what direction is the soil most variable? Why?
(d) Assume that the yield for each plot is 40 bushels in the above problem 2.
Calculate the correlation coefficient by the analysis of variance.
5 Some yields of wheat plots of a single variety grown in 10 by 10foot plots were
as follows: (Pat a from Montgomery) .
67
So
70
76
76
09
59
74
73
71
00
71
61
60
67
65
66
77
79
77
66
57
62
72
54
61
6k
80
76
76
64
68
65
67
76
fl
68
72
77
or.
lh
58
62
68
77
7H
77
70
65
oc
58
60
71
64
70
6k
65
57
75
Ik
73
63
62
69
86
6k
66
61
62
62
57
37
56
65
69
6h
63
57
61
65
58
^
73
73
71
73
71+
66
67
(9
59
53
60
78
78
73
72
73
60
7>+
Calculate the heterogeneity coefficient (intraclass correlation) by the analysis
ef varian.ce for a 2 by 1fold combination (2 horizontal rows and 1 vertical row) .
CHAPTER XIII
SIZE, SHAPE AND NATURE OF PLOTS
I. Early Use of Field Plots
Modern field experiments "began in 183^ when Jean Boussingault started a series of
tests on his farm near Bechelbronne in Alsace. Early agriculture investigators
favored large plots because of their attempts to conduct field trials in essentially
the same manner as the farmer handled his crops. '
The size of plot was considered at the Virginia Experiment Station as early as 1890
by Alwood and Price (1890) who suggested that, within limits, the larger the. plot the
more reliable the results. However, they conceded that small plots were sufficiently
accurate for preliminary trials and for obtaining information on earliness and gener
al quality of varieties. Taylor (I908) found a wide variation in size of plots used
in this country in 1908. They varied from two acres in a Georgia cotton experiment
to l/^0 acre in size, with all sizes between the two extremes. The average size of
plot in America at that time was l/lOacre .
The size of plots in relation to the experimental error was first studied at the
Rothamsted Experimental Station in 1910 by Mercer and Hall (1911). As a result of
their work and that carried on subsequently by others, the trend has been toward
smaller plots and increased replication. A questionnaire, sent out by the Committee
on the Standardization of Field Experiments of the American Society of Agronomy in
1913, reflected this tendency. The plot sizes used by different agronomists /aried
in size from one acre to l/200acre, with very few using plots larger than l/lOacre
or less than l/80acre.
At the present time, plot sizes vary from l/lO to l/lOOOacre in size. The basis for
the smaller plots with increased replication has been data from various blank or uni
formity trials conducted by Mercer and Hall (1911), Bay (1920), Summerby (1925),
McClelland (1926), Wiebe (1935), Smith (1938) and many others. The catalogue by
Cochran (1937) should be consulted for uniformity trials with specific crops. •
A — Size and Shape of Plots
II. Factors that Influence Plot Size
There are several factors to consider in plot size aside from the accuracy of the
results. Some of these are: Kind of crop, number of varieties or treatments, kind
of machinery to be used on them, and the amount of land, labor, and funds available
for the tests. (1) Kind of Crop : It is the general practice to use larger plots for
corn, sugar beets, and the forage plants than for small grains. The plots must be
large enough to carry a representative population of the crop involved. (2) Numb er
of Vari e ties or Treatments : Small plots are a necessity when large numbers of varie
ties or strains are in various testing stages. In small grains, it is not uncommon
to have from 500 to 20,000 strains in the various stages of a breeding program.
(3) Amount of Seed : In the early years of selection in small grains and in many
other plants, only a very small amount of seed is usually available. Obviously, the
plots must not be too large for the seed supply, (k) Kind of Machinery : The area
and shape of field plots should be such as to enable the operation of standard farm
machinery and to reduce to a reasonable minimum the errors concerned therewith.
Larger plots are necessary when the crop is planted, cultivated, and harvested with
standard farm machinery than where hand methods are used. (5) Land Area: For a
given area of land, the plot size varies inversely with the number of varieties or
treatments to be included. This is true until the minimum practical size is reached.
As a result, to quote Goulden (1929)'. "The general practice is to use quite small
plots adequately replicated for strain tests, i.e.. when there are a large number of
varietal units, and larger plots when the number of varieties is small enough to per
mit their use with the amount of land available." (6) Funds Available : In general,
it is more costly bo use large plots than siaa.ll plots.
III. Kinds of Experimental Pl ots
It is necessary to distinguish between nursery and field plots more or less arbitrari
ly. Nursery plots are usually • small plots cared for by hand while field plots are
larger' and adapted to the use of standard farm machinery. The present tendency is to
reduce the size of field plots and to enlarge nursery plots from single to multiple
short rows (rodrows in many cases) .
(a) Nursery Plots
Nursery plots may be as small as one square yard in area, but the rodrow is
probably the most common unit size. Small plots allow the preliminary testing of
many strains. However, uniform soil and careful t'echnic is vital to accuracy for
small plots. Taylor (I908) points out that small mistakes on small plots may greatly
modify the results. For example, an error of 5 pounds on a 1/20acre plot would mean
an error of 100 pounds on an acre basis. The rodrow unit has been widely used in
this country for small grain trials while the chessboard plot has been used in Eng
land. Engledow and Yule (I.926) describe the latter as. being one yard square with the
crop spaceplanted at 2 by 6 inches. The principal objection to the chessboard is
the amount of detailed hand labor involved and the fact that it affords less oppor
tunity to observe strength of straw, evenness of germination, etc. As plant individ
uality must be considered in row crops, there is some variation in type of nursery
plots.
(b) Field Plo ts
For standard farm machinery, field. plots usually vary from 1/10 to 1/100 acre
in size. They offer more opportunity to observe crop behavior under conditions com
parable to those found on the farm. Field plots are used for variety tests, crop
rotation experiments, fertilizer trials, forage experiments, pasture experiments,
irrigation studies, cultural trials, etc. 0rd.ina.rily, such plots are long and narrow
in shape as most convenient for farm machinery.
( c ) Co mparison of Nurse ry vs. Fi e Id PI ots
The use of small handsown nursery plots to test yields of agricultural crop
varieties has been frequently criticised, on the ground that such plots do not repre
sent normal agricultural conditions. In general, small plots have been found to com
pare favorably with large field plots in accuracy so long as adequate precautions
have been taken against competition and. other errors. There is further evidence that
nursery plots give results that are valid when applied to agricultural practice.
As early as 1910, Lyon (I9IO) reported., a comparison of seven l/l0acre plots
with seven groups of 10row plots "I" feet long. The probable errors were p.09 and.
k.k');. respectively. Moreover, less land was required for the small plots. Seven
l/lOacre plots covered an airea of 30, 1 +92 square feet,, while 70 of the 17foot rows
required only 1,190 square feet in area.
A general correspondence of rodrows and. field plots has been shown by Klages
(1933) for 11 to 1^ varieties of spring wheat, 7 varieties of durum wheat, 12 to 15
varieties of oats, 13 to 20 varieties of barley, and 7 varieties of flax in each of
k years. He calculated the correlation, coefficients (r) for the two sets of plots.
Hayes and others (1932) compared, the yields of 16 wheat varieties sown in rod rows by
1^3
hand and by a drill at different rates with those aown by a farm drill in l/UOacre
plots. . The correlation coefficients indicate some agreement between the yields ob
tained from the small and large plots. Smith (1936) has criticized the correlation
coefficient as inefficient in such comparisons: "If real differences between varie
ties were either small or nonexistent, then the correlation coefficients would be
zero or insignificant, altho the trials might agree in showing no significant differ
ences between them. On the other hand, the correlation coefficient could not become
unity unless experimental error could be entirely eliminated. Consequently, r may
vary from to + 1 even while the two forms of trial are in perfect agreement."
In a study of 12 timothy varieties, Smith and Myers (193*0 showed that the
yields from rodrows and l/50acre field plots agreed to precisely the degree required
by statistical theory. Smith (193^) later compared 9 wheat varieties sown by a farm
drill in l/lOOacre plots and dibbed in square yard plots. Agreement of the two ex
periments was excellent with respect to yield of grain. Tysdal and Kiesselbach (1939)
compared 2 varieties of alfalfa in l/30acre, field plots with various l6foot nursery
plots which differed as to number of rows and spacing. They combined the forage
yields into a single analysis of variance from which they concluded that the several
types of nursery plots gave essentially the same yields of the two varieties as did
the field plots. The interaction of varieties x type of plot was not significant.
The problem resolves itself into whether small nursery plots with more pre
cise control of soil heterogeneity will give the same results as large field plots
with less control of soil variability. The sacrifice in plot (sample) size must be
balanced l>j more effective control of soil heterogeneity for the small nursery plot
to be ae satisfactory as the large field plot. This can be brought about to some ex
tent by increased replication of small plots.
IV. Relation of Plot Size to Accuracy
In general, it has been found that the variability is decreased as the plot is in
creased in size up to about l/lOacre. However, the variability is less when a unit
of a certain area is made up of several distributed units than when a single large
unit is used. In a theoretical discussion, Siao (1935) states "Increasing the size
of plot decreases the variability of the experiment by increasing the precision of a
single plot yield. On the other hand, there is an increase in the variability within
the block through expanding the area included in the block. There are two opposing
tendencies that affect the experimental error as the plot changes in size, the final
result being due to a balance between these two tendencies. The slow rate of reduc
tion in experimental error through increase in size of plot and, in exceptional cases,
the greater variability for larger plots, may be explained by increase in variation
within the block as the plot increases in size." The work of Stadler (1921) and
Wiebe (1935) indicates that the total variation tends to increase as more land is
added to the experimental area, provided the size and shape of the ultimate unite
remains the same. It should be emphasized that plot size varies with the conditions
of the experiment, there being no one size best for all crops on all soils. Compara
tive studies on plot size have been carried out in most instances on blank or unifor
mity tests. After optimum plot size has been determined, the standard error per plot
and the number of replications to reach a given degree of accuracy in the comparison
of the mean treatment yields is usually computed. Typical investigations on plot
size will be considered for small grains and for other crops separately.
(a) Small Grain Plots
Much of the earlier work was conducted with small grains . The conclusions
applicable to one are generally .applicable to the others. Mercer and Hall (19H)
used uniformity trial data for an acre of wheat, the field being divided into 500
small plots each of which was harvested, separately. Adjacent plots were grouped so
Ikk
as to form plots of different sizes. The standard deviations in per cent p or the
1/500, 1/250, 1/125, l/ l 30, 1/25, and l/lO.acre plots were 11.6., 10.0, 8.0, 6.5., p,7,
and 5l.j respectively. The standard deviation was reduced as the plots were ioade
larger, hut the increase in plot size above l/50acre produced a relatively small
decrease in variability. These investigators found that precision was increased more
rapidly by replication. When five scattered l/300»acre plots were combined so as to
give a total area of l/lOOacre the standard deviation in per cent of the mean was
reduced to k.S per cent. Olmstead (IQl't) found with wheat that a number of small
plots ranging down to 0.0007acre in size is much better than the same total area in
one plot, and also that one large plot is more accurate than one small one. In wheat
studies, Day (1920) found that the probable error decreased with an increase in plot
size up to l/20acre. From a blank
.i.ax
with wheat, Smith (193$) concluded that the
reduction in variability by increasing plot size
equivalent random renlication.
less than could 'be obtained by
Hayes (I923) compared 16 and 32 foot rows of wheat, oats, and barley. He failed to
find a significant difference in favor of 32foot rows. A comparison of one and two
rodrows plots indicated little advantage for the harvest of two rodrow per plot
over one. Stadler (1921) obtained data on three and fiverow plots, the border rows
being discarded. His results follow:
Crop
Ho. Plots
Coefficient of Variability
One Central How Three Central Row
Barley
Oats
Wheat
21
20
On
2^.80
oh Po
27.68
22.13
22.59
25.ll
Summerby (1925) found very little difference in accuracy between large and small
plots when eight replications were used. His oat plots were 1,2,^,8,16, and. 32 rows
in width, spaced one foot, and 15 feet long. Love end. Craig (1938) made an analysis
of data from 2 oat crops for various types of plots and various numbers of replica
tions, and for rows 15 and 30 feet in length. The data indicate that ^row plots
with several replications (8 or 10), when all rows are harvested, give accurate re
sults. They are preferred to single row plots. The 15foot rows were considered
more satisfactory than those 30 feet in length. Such data as these support the wide
spread practice of using three rodrow plots for small grain nursery trials with the
center row harvested for yield.
(b) Other Crops
Different crop plants are known to differ in variability. The coefficients
of variability for different crops were compared by Smith (1933) for a standard
l/^0acre plot from the published data for 39 uniformity trials. The crops fell
roughly into 3 groups: (I) wheat, mangolds, sugar beets, soybeans, and sorghums
(forage) seem to be less variable: (2) corn, potatoes, cotton, and natural pasture
were intermediate; and (3) fruit trees were most variable.
In the ca.se of corn, Bryan (1933) reports that "variability of plot yields
decreased as the size of plots increased from 3 to lb, to 2k } and to kS hills, but
the decrease was not proportional to the size of plot , The experimental error for a
given area, therefore, would be lower with larger numbers of small plots." McClel
land (1926) obtained a similar reduction in error as the size of plot was increased,
the error being 11.2 per cent for l/80acre plots, and 6.2 per cent for those 1/2
acre in size.
With sorghums, Stephens and v r inall (1928) concluded that the errors decrease
with an increase in plot size up to l/20acre. Increasing the plot from l/oOOacre
1*5
to l/20acre, with the same total area concerned, reduced the probable error about
60 per cent.
The standard error was found "by Immer (1933) to "be actually reduced in sugar
"beet plots when the plot size was increased from one to two rows in width, or for an
increase in length from two to four rods. However, efficiency in the use of land de
creased as the size of plot was increased. Some of his data for the harvest of the
entire plot are as follows :
Length Plot Percentage Efficiency of Plots of Indicated Width (Rows)
in Pods 1 2 3 ^ 6 12
2
100.0
88.0
77.7
53.3
3^9
21. h
k
76.2
62.5
kB.2
35.2
21.2
28.8
10
50.0
37.6
28.6
26.1
10.2
9A
20
35.1
2*. 5
21.6
10.1
5.8
6.7
Similar results were reported by Immer and Raleigh (1933)«
Uniformity trial data with soybeans, computed by Odland and Garber (1928),
indicate that 16foot plots in single rows replicated three times were the most satis
factory when both accuracy in results and land economy were taken into account.
Vest over (192*0 experimented with 220 rows of potatoes, 150 feet long. He
harvested them in 10foot lengths, and found a sharp reduction in probable error
between row lengths of 10 and Uo feet. Beyond 60foct lengths, there was very little
reduction in error.
Ligon (1930) found no necessity for rows greater than 100 feet in length for
cotton, the shorter rows being just as accurate when sufficiently replicated. Unit
rows of cotton 2k feet long and spaced one foot apart were used by Siao (1935) in
studies on size of plot for cotton. When combined into plot sizes of 1,2, 3 } k, and 3
rows, the efficiency was greatest for the smallest plot.
In plot size studies with millet, Li and others (193&) concluded that plots
15 feet long and two rows wide were the most efficient, i.e., 1139 per eent compared
to 100 per cent for 15foot plots one row wide.
Batchelor and Reed (1918) studied the variability of orchard plot yields from
the standpoint of increasing the number of adjacent trees per plot. The average re
duction in variability for all fruits was 37.78 to 2^.27 per cent when the plot was
increased from one to eight trees, but little was gained by including 16 to 2k trees
per plot .
The reasons for variability in small plots may be summarized as follows:
(1) Variability in soil, (2) losses in harvest and errors in measurement have a rela
tively great effect, (3) in row crops, plant variability may be important because of
fewer plants, (k) competition and border effects are apt to be greater on small plots.
V. Plot Sizes for Various Crops
The plot sizes depend upon the crop plant, and upon the conditions under which the
test is conducted. (1) Small Grains : The majority of experiment stations use three
row plots with the center row harvested for yield, but a few use fiverow plots with
the center three rows used for the yield determination. A few use singlerodrow
plots. (2) Corn : The Nebraska station uses fourrow plots, 12 hills long, harvesting
the center rows for yield. Others use single rows about 20 hills long, or three rows
146
of the same length with only the center one harvested for yield. Bryan (1933) re ~
ports that,, in a comparison of openpollinated varieties and hybrids,, equal degrees
of precision were attained with about half as many plants or hills of crosses as of
openpollinated varieties. He found, that '43 total hills were sufficient to represent
a variety. (3) Soybeans : Soybeans may be grown in rows lo feet long and 30 to 32
inches apart. Field plots are often employed, (4) Sorghu ms : The work of Stephens
and Vinall (1923). indicates that "three 02' four replications of l/40acre or l/80
acre plots will give results sufficiently reliable for the ordinary sorghum test".
Slightly larger plots are advocated by Swans on (1930). When protected by borders,
2 and 4row plots 8 rods long having an area from l/pO to I/25acre, are regarded as
convenient units. At Kansas, four row plots about 100 feet long are used. The grain
sorghums are thinned to eight inches in the row., while the forage sorghums are spaced
four inches in the row. The rows are spaced the same distances apart as for corn.
(5) Alfalfa and Clove rs: These crops are usually grown in field plots about seven
feet wide and 60 feet or longer in length, with the center five feet harvested with
a mower. Tysdal and Kiesselbach (1939) state that the most serviceable types of plot
for advanced nrrsery testing appear somewhat optional among these: (a) Soliddrilled
5 to 8 rows sioaced 7 inches apart with a 12 to 14inch alley between border rows; or
(b.) . soliddrilled 3 bo 5 rows spaced 12 inches apart with an 13inch alley between
border rows. The entire plot may be harvested since very little error due to border
effect occurs. (c) Single rows spaced 18 to 24 inches apart are permissibile for
preliminary nursery tests. (6) Sugar Be ets: Immer (1932) states that fourrow plots
are the most efficient. The rows should be two to four rods long, spaced 20 to 22
inches apart, and the plants thinned to about 12 inches in the row.
VI . Re lation of Sh ape _to Bel lability
Some investigators have found that long narrow plots best overcome the effects of
soil heterogeneity, while others believe that plots should be approximately square.
For example, Barber (1914) reported that a small square plot affords a more accurate
basis for variety comparisons than a long narrow plot that has extra growth along the
borders when alleys exist between the plots. On the other hand, Kiesselbach (1918)
showed, that the coefficient of variability for l/lOacre oat plots 43 rods by 5*5
feet was 3«84 per cent, as compared with p.lS'per cent for plots lo rods by I0.3 feet.
Justessn (.1932) found long narrow plots to be more efficient than the shorter plots
of the same area. Mercer and Hall ( 1911 ) divided the plots of a single variety into
plots of equal area but of different shapes. The dimensions were 20 by 12, and ;30 by
5 yards. They found no significant difference in variability between them. Similar
results were obtained by Stephens and Vinall (1928) with sorghums. Bryan (1930)* in
his work with various shapes of corn plots, concluded that shape is less important
as the size of plot is reduced;. With plots as small as lo hills, either single, two
or fourrow plots may be expected to give similar results.
These, apparent inconsistencies are explained in the work of Day (1920) who harvested.,
in fivefoot sections, a l/40acre area uniformly cropped to wheat and combined the
ultimate units to form plots of various shapes. He found that plots with their
greatest dimensions in the direction of the least soil variation are more variable
than plots having their greatest dimension in the direction of the greatest variation.
He found that shape exerted no influence on accuracy where soil variation is as great
in one direction as it is in the other. Some of his data follow:
Adjacent
Bows
TnoTT
1
3
10
24
Length
Bows
TfTTT
150
30
15
5
Total Length
Bo ws i n Blot.
(Ft7T~"
130
130
130
120
Shape of Plot
C.V
Long in direction of least 17 O^
variation
Long in direction of least to. 37
variation
Be ct angular 12 . 72
Long in direction of most 10. 3^
variation
147
Similar conclusions were obtained "by Siao (1939) for cotton and by Smith (1938) for
wheat .
It is generally conceded that relatively long and narrow plots with the long dimen
sion in the direction of the greatest soil variation best overcome the affects of
soil heterogeneity. In addition, linear plots are more economical for cultural
operations. However, the area occupied by a single replicate or block should approach
a square in shape for the most efficient design.
VII. P ractical Considerations in Plot Shape
Width of plots should be sufficient to a] low for the removal of border rows when
this appears desirable, or to render border effects negligible when not removed. The
triple rod row is a convenient shape for small plots, while large plots are usually
rectangular in shape to accommodate farm machinery in an attempt to simulate farm
conditions.
(a) Adaptation to Farm Machinery
Some multiple of seven feet provides a favorable width for field experiments
as it permits convenient operations of the 3«5 an& 7.0foot farm implements. Kiessel
bach (1928) calls attention to the fact that the multiple of seven feet will enable
the use of the sevenfoot disk,, sevenfoot drill, sevenfoot binder, and 3.5foot
corn planter and cultivator. The standards of the American Society of Agronomy (1933)
recommend 14 feet as a minimum plot width for crop rotation, fertilizer, and tillage
experiments, while varietal tests with intertilled crops commonly should contain at
least three or four rows. Extremely narrow plots, in the case of manurial or fertil
izer tests, make it difficult to keep the treatments within the plot limits.
(b) Calculation of Pl ot Size
Plots should be made an alequot part of an acre, e.g., l/40, 1/50, l/SO^acre
plots. This sort of plan is worthwhile because of the grave possibility of error in
computations made on acres expressed as decimal fractions. For instance, to calcu
late the dimensions of a l/40acre plot for a drill seven feet wide, the steps are as
follows:
43,56o/4o = IO89 square feet in l/40 acre.
1,089/7 = 155.6 feet for length of the plot.
Hayes (1923) suggests for small grain nursery rows spaced. 12 inches apart, that the
row length be adjusted in length slightly so that gram yields per plot can be con
verted to bushels per acre by multiplying by a simple conversion factor. The
factor 0.2 can be used for a 15foot row of oats, the factor 0.1 for a lofoot row
of wheat, and the factor 0.1 for a 20foot row of barley.
VIII. Calculation of Plot Efficiency
Some uniformity data on 120 rod rows of Haynes Bluestem wheat in bushels per acre, as
given by Hayes and Garber (1927), will be used for the computation of plot efficiency.
The method was suggested by Dr. F. E. limner.
25.0
24.9
29.4
28.7
28.0
27.0
28.9
20.9
25.3
24.1
, 22.0,
21.7,
, 25.0,
20.7,
, 2^. 4,
23.1,
, 26.8,
25.2,
, 23.7,
26.4,
, 23.8,
24.6,
> 27.5,
25.2,
» 27.2,
26.0,
7 27. 1+,
25.2,
, 25.8,
22.1,
22.0,
24.5,
30.7,
29.6,
27.8,
317,
27.9,
30.0,
21.6
26.9,
18.6
26.3
25.7
25.4
23.8
28.0
28.2
22.7
28.3
27.6
, 23.5,
20.3,
, 25.2,
25.6,
, 21.9,
23.2,
, 26.5,
24.4,
, 23. 1,
28.3,
, 28.1,
24.5,
, 25.4,
25.8,
, 21.6,
19.3,
, 25.1,
21.6,
, 27.6,
27.3,
19.9,
23.1,
23.2,
24.1,
28.8,
26.7,
28.1,
24.0
24.6;
30.2,
24.9
28.5
21.5
29.3
21.0
31.6
32.0
28.8
25.9
22 .4
, 22.9,
22.9,
, 27.0,
26.6,
, 25.6,
24.4,
, 24.3,
28.9,
, 23.8,
24.8,
, 23.7,
25.0,
, 29.6,
28.3,
, 25.2,
26.0,
> 24.8;
26.9,
, 23.7,
23.1,
25.0,
25.9,
28.2,
25.5,
22.8,
33.0,
27.6,
25.9,
26.5.
148
It is assumed that JO varieties are to be tested, and that the investigator desires
to determine the relative efficiency of 1 ; 2, and 4row plots. The analysis of
variance will he used.
( J. ) One Bo w per Plot :
Block S(>: b ) S(x 2 b )
T 727.2 528,819.84
II 767.. 6 089,209.70
III 326.7 603,^32.89
IV l60 JB_ ^7 3,3 36, 64
Totals 3082.3 2,380,279.13
S(x) for all plots = 3,082.30 £ = 25.685833.
S(x 2 ) for all plots  80,176.37. S(::) 2 /k ■ 79,171 J+306
S(x 2 )  S(x) 2 /N = 1,004.9394
s(x 2 b )  (sx)2 = 79/342.64  79,171.43 = 171.21
• 30 N
Variation Sum Mean
due to D. F. Squares Square
Blocks 3 171.21 37 .0690
Varieties and Error llo 833 .Jo _ 7.IS73
Total H9" 1004.94
( 2 ) Two Rows per Plot : '
Block S(xJ S(x 2 )
_ L..b „ , r : JbL._
I 1494.8 2,234,427.04
n 15§Ll5 2,520,156.23
Totals '    jogg^ 4.,754,583"29
S I : \_2  S(x) 2 = 159,661.25  79,171.43 = 659.19
2 H ™ 2
The total S(x 2 ) is divided by 2 to place the' results on a single plot oasis so
■ that the common correction factor S (::•;:) 2 /]R , can he used.
S (x2 b )  s(x)2 a 79,243.05  79,171.4;
60 N
71 .fsP
Variation Sum Mean
due to D. F, Squares Square
B 1 o cks 1 7 1 . o2 7 1 . 62
Varieties and Error 58 587.98 10. 13
(3) Your Bcvs P er Blot:
s(x 2 )  s(x) 2  3 18, 950 . 59 ,  79,171.43  561.21
4 N 4
Variation , Sum Mean
due to I). F . Squares Square
Total ' 29™ ~" ~ " 56iT21 " ~~ '"19.3921
1^9
(k) Comparison of 1 , 2, and k Sow Plots
No.
Determinations
Rows
per Plot
No.
Blocks
Mean
Square
Mean Square
Basle One Plot
Pet.
Efficiency
30
50
50
1
2
k
It
2
1
7.1875
10.1307
19.5521
7.1875
5.0655
I4..838O
100.00
70.95
37.1^
U. Replication
B — Plot
in Experimental Work
Replication
Replication is merely repetition. The investigator repeats a variety or treatment
several times in a test in order to obtain a moan yield or value which is a more
reliable estimate of the yield of the general population than that obtained from a
single plot of a treatment. It also provides the mechanism for a valid estimate of
the random errors in an experiment . Strictly speaking, five replications of a
variety refer to six plots, i.e., the original plot and its repetition five times.
For the sake of simplicity, the number of replications will be understood to mean the
number of plots grown of each variety or treatment. In field experiments, a single
replicate is usually planned to contain one plot of each treatment in a rather com
pact block. The repetition of the treatments is brought about by the repetition of
the blocks. This distribution of plots over the experimental area is an effort to
sample" the field in an attempt to measure and, in some cases remove, the influence
of soil heterogeneity. Replication in space and time is often necessary. For exam
ple, it may be desirable to repeat an experiment in other regions of the state in
order to sample different soil and climatic conditions. In the same region, repeti
tion of the experiment over a number of years may be necessary to sample the climatic
conditions in different seasons.
X. History of Replication
Replication of experimental plots has been comparatively recent. In the old field
tests large single plots were placed side by side. These were simple and effective
for the demonstration of known facts so long as the differences to be observed were
large. However, they are inadequate as soon as accurate measurements are needed be
cause they do not take into account the tremendous variation in the soil from plot to
plot.
Sir John Russell (1931) gives some of the early history of replication. The Broad
balk plots at the Rothamsted Experimental Station were split lengthwise into two
halves in 13^61+7 which, from that time onwards, were harvested separately. This was
the first duplication of field experiments so far as can be determined. In 18^7^8,
and occasionally afterwards, one half of each plot was treated differently from the
other with the result that they ceased to be strict duplicates. Better duplication
appears to have been practiced by P. Nielsen, founder of the Danish. Experiment Sta
tion about 1870, in his experiments on grass mixtures for pastures. Some Norfolk
(England) experiments carried out in the later l880's were systematically replicated
as follows: ABCDDCBA. In America, some experimenters began to use replication about
1888. Some old experiments in Kansas were replicated six times. However, replica
tion soon fell into disuse because of the demand for information and due to limited
land and funds. Single plots wore the rule. Nothing further was done in England
until 1909 when A. D. Hall (1909) and later Wood (1911) urged the need for the esti
mation of experimental errors. Marked changes came about as a result. S. C. Salmon
(1913) revived duplication of plots in this country in 1910. Single l/lOacre plots
were commonly used for variety and rate and date of seeding tests with small grains
150
at that time. He split those l/lOacre plots into l/50acre plots and replicated
them five times for variety tests. His rate and date tests were replicated three
times. Thus, the same area was required for variety tests e„s "before and a smaller
area for rate and date tests. Largely as the result of his effort s, the Office of
Cereal Investigations, U. S. D. A., provided for replication in their work about
1912. In England at about the same time, Dr. E. S. Beaven designed his wellknown
strip method of replication which is especially suited to variety trials.
A questionnaire sent out by the Committee on Standardization of Field Experiments of
the American Society of Agronomy in I918 indicated that less than 20 per cent of the
agronomic workers depended iipon single plot tests even though they had been the rule
10 years previously. At present, replication is considered essential in modern field
experiments.
XI . Reduction o f Error by Replication . .
The most effective method to obtain greater accuracy in field experiments as well as
in many other types of agronomic experiments, is to increase the number of replica
tions. It can be brought about to a limited extent ''oj an increase in plot size as
shown by Summerby (1923). However, frequent' replication of small plots^proved to be
a more efficient means to obtain a high degree of accuracy than the use of the same .
amount of lend with less frequently rer>licated larger plots. Love (1936) gives some
uniformity trial data with cotton that indicate the same trend. The probable error
for a 2 row plot 20 feet long was 10.35 per cent, while that for two single 20foot
plots was 9.01. Further, the probable error for a single kvow plot was 9,51 while
for the same area made up of four scattered units it was 755 P e ** cent. Many other
investigators have obtained similar results. Since the standard error of the mean
(o£) is given by 05; = s , it follows that the decrease in Oj? is proportional to
7 1?"""
the square root of the number of replications. This rule applies when the variation
due to the replicates themselves is removed from the error, but not strictly other
wise. This can be illustrated with some data on 120 rod rows of bluestem wheat,
cited by Hayes and Garber (1927) • The value of replication was studied on the varia
bility of yields calculated separately on the basis of 20 determinations and for
1,2,^, and 6 systematically distributed plots. The coefficients of variability were
compared with mathematical expectation as follows:
No . No . sy st emat I cally Mathemat leal
determinations distributed plots C. V. expectation
20 I
20 2
20 k
20 6
9.05
9.05
S..3h
9.03A/2 =
= 6.1+2
5.61
j + . 53
k.hk
369
The calculated, coefficient of variability  decreases as a result of replication, but
less rapidly than would be indicated, by mathematical expectation. This is attributed
to the greater land area used for several replications than can be used for single
plot trials which, on the average, brings in soils of greater difference in producti
vity than can be found in smaller areas. In this case, the error due to blocks has
not been removed. That replication beyond a certain point may be impractical Is in
dicated in some data compiled by Salmon (1923) . He shows the relation between the
number of replications and the probable error of the mean (expressed In per cent) as
follows :
Number of Plots 1 2 3 k / 5 0789
Kherson oats
3.7
2.0
2.0
1.7
1.0
1.8
1.7
1.6
1.5
Alfalfa
11.2
713
71
5.0
k.Q
M
5.5
6.0
5.8
Ear corn
90
5.9
5.5
5.1
k.l
37
k.l
k.l
5.3
151
It is to be noted that variability was rapidly reduced up to k replications, but the
decrease was at a much slower rate beyond that point. Hayes and Garber (192?) ques
tioned whether the gain in accuracy beyond three replications warranted the addition
al work. The relation of replication to design will be considered in a later chap
ter.
XII. Number of Replications
The question naturally arises as to the number of replications that should be used.
Goulden (1929) states that it depends upon the degree of soil heterogeneity, the
degree of precision required, and the amount of seed available. Any desired degree
of precision within practical limits may be ordinarily achieved for any given set of
conditions by replication. For field plots, the American Society of Agronomy (1953)
recommends 3 to 6 replications, dependent upon the degree of precision required. The
smaller number will suffice when average rather than annual results are stressed.
From k to 6 replications are commonly used in corn variety trials. Nursery experi
ments ordinarily should be replicated 5 "to 10 times to assure significant results.
It is impossible to prescribe a rule for all cases. In rodrow trials with oats,
Love and Craig (1938) found 8 or 10 replications more satisfactory than a smaller
number, as 3 or 5. In alfalfa nursery plots, Tysdal and Kiesselbach (1939) concluded
that k to 1.6 replications were necessary to make a 5 per cent difference statistical
ly significant for plots that varied in size from l/80acre to a single spaceplanted
16foot row. The larger plot required the k replications. However, little is to be
gained by the use of more than 10 replications in field trials.
References
1. Alwood, W. B., and Price, R. H. Suggestions Regarding Size of Plots. Va. Agr.
Exp. Bui. 6. 1390.
2. Barber, C. W. Note on the Influence of Shape and Size of Plots in Tests of
Varieties of Grain. Me. Agr. Exp. Sta. Bui. 226, pp. 768!+.. 191U.
3. Batchelor, L. D., and Reed, H. S. Relation of the Variability of Yields of
Fruit Trees to the Accuracy of Field Trials. Jour. Agr. Res., 12:2^5283.
1918.
h. Bryan, A. A. A Statistical Study of the Relation of Size and Shape of Plot and
Number of Replications to Precision in Yield Comparisons with Corn. la. Agr.
Exp. Sta. Rpt. for I9303I, p. 67. 1931. '
5. . Factors Affecting Experimental Error in Field Plot Tests with
Corn. la. Agr. Exp. Sta. Bui. 163. 1933.
6. Cochran, W. G. Catalogue of Uniformity Trial Data. Suppl • Jour. Roy. Stat.
Soc, U:233253 1937.
7. Day, J. W. The Relation of Size, Shaoe, and Number of Replications of Plots to
Probable Error in Field Experiments. Jour. Am. Soc. Agron., 12:100106. 1920.
8. Engledow, F. L., and Yule, G. U. The Principles and Practices of Yield Trials.
Emp. Cotton Growing Corp. 1926.
9. Goulden, C. H. Statistical Methods in Agronomic Research. Can. Seed Growers
Assn. 1929.
10. Hall, A. D. The Experimental Error in Field Trials. Jour. Bd. Agr. (London),
16:365370. 1909.
11. Hayes, H. E, Controlling Experimental Error in Nursery Trials. Jour. > Am. Soc.
Agron., 15:177192. 1923/
12. Hayes, H. K., and Garber, R. J. Breeding Crop Plants. 'McGrawHill. pp. 6977.
1927.
13* — et al. An Experimental Study of the Rod Row Method with Spring
Wheat. Jour. Am. Soc. Agron., 2^:950960. 1932.
152
Ik . Immer, F. R. Size and Shape of Plot In Relation to Field Experiments with Sugar
Beets. Jour. Agr. 'Res., 1+k: 61+9668. 1932.
15. and Raleigh, S. M. Further Studies of Size and Shape of Plot in
Relation to Field Experiments with Sugar Beats. Jour. Agr. Res., 1+7:591598.
1933.
16.. Justensen, S. H. influence of Size and Shape of Plots on the Precision of Field
Experiments with Potatoes. Jour. Agr. Sci., 22:366372. 1932.
17. Kiesselbach, T. A.. The Mechanical Procedure of Field Experimentation. Jour. Am.
Soc. Agron., 20:^33^2. ' 1928.
18. Studies Concerning the Elimination of Experimental Error in
Comparative Crop Tests. Nebr. Agr. Exp. Sta. Res. Bui. 13 . 19l8>
19 Klages, K. H. W. The Reliability of Nursery Tests as Shown by Correlated Yields
from Nursery Rows and Field Plots. Jour. Am. Soc. Agron., 2^:k6kk'j2 . 1933.
20. Li, H. W., Meng, C. J., and Liu, T.N. Field Results in a Millet Breeding Ex
periment. Jour. Am. Soc, Agron., 28:115. 195$ •
21. Ligon, L. L. Size of Plot and Number of Replications in Field Experiments with
Cotton. Jour. Am. Soc. Agron., 22:689699. 1930.
22. Love, H. II . Statistical Methods Applied to Agricultural Research. Com. Press
Ltd. (Shanghai), pp. 1+0*+ 1+20. I.936.
23. Love, H. H., and Craig, W. T, Investigations in Plot Technic with Small Grains.
Cornell U. Memoir, ~2±k . , 1938.
2k. Lyon, T. L. A Comparison of the Error in Yield of Wheat from Plots and from
Single Rows In Multiple Series. Proc. Am. Soc. Agron., 2:3339 1910.
25. McClelland, C. K. Some Determinations of Plot Variability. Jour. Am. Soc. Agron.,
18:819823. 1926.
26. Mercer, W. B., and Hall, A. D. The Experimental Error in Field Trials. Jour.
Agr.. Sci., U: 107132. 1911.
27. Odland, T. E., and Garber, R. J. Size of Plat and Number of Replications in
Field Experiments with Soybeans. Jour. Am. Soc. Agron., 20:93108. 1928.
28. Olmsted, L. B. Some Applications of the Method of Least Squares to Agricultural
Experiments. Jour. Am. Soc. Agron., 6:19020l+. 19ll+.
29. Report of the Committee on Standardization of Field Experiments. Jour. Am. Soc.
Agron., 10:3^535!+.. 1918.
30. Report of the Committee on Standardization of Field Experiments. Jour. Am. Soc.
Agron., 25:803828. 1933.
3i. Russell, E. J. The Technique of Field Experiments (Forword) . Rot hams ted Conf.,
13, pp. 58. 1931.
32. Salmon, S. C. A Practical Method of Reducing Experimental Error in Varietal
Tests. Jour. Am. Soc. Agron., 5:l82l84. 1913.
33. Some Limitations in the Application of the Method of Least Squares
to Field Experiments . Jour. Am. Sec. Agron., 15:225229. 1923 •
3I+. Siao, Fu. Uniformity Trials with Cotton. Jour. Am. Soc. Agron., 27:97^979
1935.
35. Smith, K. Fairfield. Comparison of Agricultural and Nursery Plots in Variety
Experiments. Jour. Counc. Sci. and Ind. Res., 9:207210. 1936.
36. An Empirical Law Describing Heterogeneity' in the Yields of
Agricultural Crops. Jour. Agr. Sci., 28:123. 1938.
37.  , and Myers, C. II . A Biometrical Analysis of .Yield Trials
with Timothy Varieties using Rod Rows. Jour. Am. Soc. Agron., 26:117128.
193^ .
38.. Stadler, L. J. ■ Experiments in Field Plot Technic for the Preliminary Determina
tion of Comparative Yields in Small Grains, Mo. Agr. Exp. Sta. Res. Bui. 1+9 .
1921 ,
39. Stephens, J. C .., and Vinall, II. N. Experimental Methods and the. Probable Error
in Field Experiments with Sorghum. Jour. Agr. Res., 37:629646. 1929.
1+0. Summerby, R. Replication in Relation to Accuracy in Comparative Crop Tests,
Jour. Am. Soc. Agron., 15:192199 1923.
153
kl . Summerby, R. A Study of Size of Plats, Number of Replications, and the Frequency
and Methods of Using Check Plats, in Relation to Accuracy in Field Experiments.
Jour. Am. Soc. Agron., l^:lU01^9. 1925.
k2 , Swanson, A. F. Variability of Grain Sorghum Yields as Influenced "by Size, Shape,
and Number of Plots. Jour. Am. Soc. Agron., 22:833838. 1930.
kj. Taylor. F. V. The Size of Experiment Plots for Field Crops. Proc. Am. Soc.
Agron., 1:5658. 1908.
bk. Tysdal, H. M., and Kiesselbach, T. A. Alfalfa Nursery Technic. Jour. Am. Soc.
Agron. 31:8393. 1939.
k5 . Vestover, K. C. The Influence of Plat Size and Replication on Experimental
Error in Field Trials with Potatoes. W. Va. Agr. Exp. Sta. Bui. 189. 192^.
^6. Wiebe, G. A. Variation and Correlation in Grain Yield Among 1500 Wheat Nursery
Plots. Jour. Agr. Res., 50:331357 1935*
b'J . Wood, T. B. The Interpretation of Experimental Results. Jour. Bd. Agr. (London)
Supplement, pp. 1537. 1911.
Questions for Discussion
1. What was the early history of the use of field plots?
2. What type of experiment is used to compare different sizes and shapes of plots?
Why?
3. What practical considerations usually determine the size of plots used in field
experiments?
b. What is the general objective of nursery tests and what general relation should
they have to field plots?
5. Distinguish between nursery and field plots.
6. What are the common sizes of nursery plots? Field plots?
7. Compare nursery plots and field plots as to accuracy.
8. What is the relation between size of plots and the standard error? Between size
of plot and border effect?
9. How may increased size of plot increase the amount of variability?
10. What size of plot has shown the lowest variability for practical purposes with
wheat? Corn? Soybeans? Millet? Cotton?
11. What is meant by efficiency in plot size?
12. What reasons can be given for the variability in results with small plots?
13. What is a common size of plots for small grain nurseries? Corn trials? Sorghums?
Alfalfa? Sugar beets?
Ik. In general, what relation is found between shape of plots and the standard error?
15. What relation, if any, is found between the direction in which plots extend and
the standard error? Why?
16. What recommendations would you make on width of field plots for the use of farm
machinery? Why?
17. What relation is found between shape of plots and border effect?
18. What modifications can be made in length of nursery rows for wheat, oats, and
barley, for rapid conversion of yields to bushels per acre?
19. What is replication? Why used?
20. What has be«*s the general practice regarding replication of plots? What is the
practice now?
21. Trace the early history of plot replication.
22. What serious results may result from single (unreplicated) plot trials? Why?
23. What is the theoretical relation between the number of plots and the standard
error? Actual relation? Why do they not always agree?
2b. What class of errors does plot replication tend to reduce or eliminate? On what
class does it have no effect?
23. Diccuss the statement: "Precision can be' increased Indefinitely by replication."
26. How does replication furnish an estimate of error'
27. Give a general rule or rules for plot replication.
Problems
1. It is desired to use l/30acre plots in a crop rotation experiment and to make
them lh feet wide. Calculate the plot length.
2. Gome data reported by Wiebe (1935) are given below for 15foot rows of wheat; one
foot apart. The yields are reported in grams. Assume I5 varisties, and compute
the efficiency for 1, 2, and 4 row plots.
Series 2 Series 3 Series k
Scries 1
715
770
76O
663
753
7^5
6hi
7.85
360
685
755
61+0
725
715
700
595 380 580
710 655 67.5
715 690 690
613 685 353
730 670 380
670 585 560
690 530 520
1+93 ^55 [ '70
5I+0 lj50 300
730 610 500
810 665 570
635 585 ^63
C53 530 ■ l r55
773 615 5^5
705 355 Mo
5. Calculate the number of replications required to make a 5 per cent difference in
yield statistically significant for these sizes of plots:
Kind of Plot 05 (in per cent)
Field plot ' 3.3
Single row plot (l'8inch spacing) 32
Single row plot (2l+inch spacing) hand planted 7.0
Use the formula, Oj (2),/ 2 = 5 (percent difference in yield?), where "n" = th«
number of 'replications.
____  _^^^^^^^^^^^^^^_
CHAPTER XIV
COMPETITION AND OTHER PLANT ERRORS
I. Plants in Relation to Error
That soil heterogeneity will contribute to experimental error has already "been seen.
There are also many errors due to plants that may contribute to the experimental
error. These may be caused by differences in genetic constitution or variations due
to environmental conditions. Variations in plant stand within plots may introduce
differential responses due to intraplot competition, while the effect of one plot
on the adjacent one may bring about differences due to inter plot competition. Other
errors related to plants include such "effects as differences in the moisture content
of the harvested crop, differences in adaptation, etc
II. Acclimatization
il serious systematic error may be introduced thru differences in acclimatization of
the crops under test, unless acclimatization itself is the factor under consideration.
Varieties in crops like corn, alfalfa, and red clover may vary widely in their clima
tic adaptation. Variety tests in corn may be a common source of error in this re
spect. In Nebraska, Kiesselbach (1922) compared Reid Yellow Dent corn grown 100
miles farther north with that grown and adapted at Lincoln. He obtained large dif
ferences in yield, plant height, date of maturity, length of ear, etc., within the
same variety when originally grown under different conditions. Lyon (19H) reported
similar results for corn and also for strains of Turkey wheat from other states in
cluded in winter hardiness tests. Differences in varieties may be brought about in
a very few years which may introduce either a slight or a very large error. Reliable
tests are impossible when varieties are collected from different climatic regions.
Each variety should be grown for a year or two in the region where it is to be test
ed until it has undergone the changes incidental to adaptation to the new environment.
III. Plant Individuality
Plant individuality varies with different crops. It is more marked in cross than in
self fertilized crops, e.g., it would be more important in rye than in barley. The
size of plot necessary is influenced by the number of plants grown per plot, as well
as by the kind of plant. For instance, it is easily possible to have 1,000,000 wheat
plants on one acre, while the number of corn plants is only about 10,000 per acre.
Plant individuality would be negligible in the case of small grains, but quite impor
tant in crops like corn and sorghums where the number of plants per plot may be quite
low. Lyon (19H) found that quite a large error may be introduced by yield deter
minations from a small number of plants due to the variations in growth of certain
individuals. For maize, he showed that the effect of plant individuality was prac
tically none when each plot was composed of 100 plants.
IV . Variation in Moisture Content of Harvested Crop
In forage and cereal crops the variation in moisture content of the harvested crop
may be an important source of error in yield determinations. For precise experimen
tal results, this condition should be recognized and a remedy provided for it.
(a) Moisture in Forage Crops
Obviously, the most accurate method to determine the water or. drymatter con
tent of the forage grown on a plot is to dry all the material to a water free basis.
Since it is impossible to do this, dry matter determinations are based on small
shrinkage s amp 1 e s .
155
156
The problem of moisture determination is rather simple under semi arid con
ditions where forage is readily fieldcured. Forage weights arc usually taken after
the material is dry enough to stack, talcing a shrinkage sample at that time. The
sample is weighed immediately and allowed to air dry for 2 or 3 weeks after which it
is reweighed for the airdry weight. Yields" corrected on this "basis are found to
he reliable. In case moisture free determinations are necessary, the samples of each
variety or treatment may be composited, ground, and dried in a vacuum oven for 12 to
2k hours .
Under humid conditions, reliable comparisons from the weights of fieldcured
forage cannot be made, except on rare occasions that cannot be predicted. As a re
sult of the work of Farrell (191*1); McKee (191k) j 'Vinall and McKee (1916), and Amy
(I9l6), the general practice has been to weigh the forage as soon as cut, and sampling
it for airdry or waterfree determinations at that time. These green samples are
usually placed in a drier" at once to avoid the loss of dry matter thru oxidation,
fermentation, etc. The investigations of Wilkins and Eyland (.1938) indicate that the
samples should be taken and weighed within k to 6 minutes after the forage is cut to
avoid error due to moisture less. These workers also found that the error introduced
thru the use of green weights of alfalfa and red clover for plot yields without dry
matter determinations was negligible so long as the weights were taken quickly.
Yield determinations on the basis of green weights proved to be as accurate as where
2 or 3 samples per plot served as a basis for moisture determination, and subsequent
yield c orr e ct i on .
(b) Moisture in Cer eals
The moisture content of small grains is usually of little consequence since
the bundles are usually airdry before threshing is attempted. The threshed grain
may be weighed at the threshing machine and re weighed a week later to be sure it
has reached a uniform moisture content. The determination of moisture in shelled
corn is regarded, as an essential practice for obtaining precise yields. Moisture
determinations can be made on each plot of each variety, or a composite determination
for the variety on all replications. A common practice is to report yields on the
basis of shelled corn with 15 • 5 per cent moisture, the maximum moisture permitted for
U.S. No. 2 corn. Moisture determinations for corn or small grains can be made quick
ly with the BrownDuvel moisture tester described oy Coleman and Boerner (1927) •
Recently, the TagHeppensta.ll moisture meter, an electrical device, .has been widely
used for moisture determinations. This meter is calibrated for wheat, corn, oats.
barley, rye, sorghums, rice, soybeans, and vetch. The electrical moisture meter
has certain advantages for practical work: (l) It is unnecessary to clean after
each sample; (2) the sample is not weighed; (3) a single determination can be made
in less than one minute; (4) it will duplicate results within a tolerance that can
not be met in a single determination by other methods, and (5) the operation and
maintenance cost is low. Cook, et al (I93V) have made a study of rapid moisture
determination devices. When determinations are made on each variety in each repli
cate, single rather than duplicate determinations should be sufficient.
V . C ompetition C oncept in Pl ants
A "struggle for existence'' results when plants are grouped or occur in communities in
such a way that the demands for an essential factor are in excess of the supply.
This is true in many field trials. Competition always occurs when two or more plants
make demands for light, nutrients or water in excess of the supply. It is greatest
between individuals of the same species which make similar demands upon the supply at
the same time. This is generally the case in cultivated crops where an area is
planted to the same species or variety. A detailed discussion on the nature of plant
competition is given by Clements and others (1929)
157
A number of investigations have "been conducted to determine the importance of plant
competition in experimental plots. There is apt to "be an effect when varieties that
differ considerably in growth habit, time of maturity, and other characters are
grown in adjacent plots. The principal contention is whether or not the yield of a
poorer variety growing next to a high yielding variety will he adversely affected so
that the yield will be actually lower than when the variety is grown next to a plot
of its own kind.
Competition may or may not influence plot yields. Two distinct schools of opinion
have arisen as to its importance. In areas of limited moisture supply, competition
has been generally found to be a source of error in comparative crop tests. Kiessel
bach (1918) obtained errors of 2k and ^6 per cents due to plant competition in two
different years. Hayes and Arny (1917) found errors in small grain yield trials
where varieties competed with each other. In Missouri, Stadler (1921) reported errors
of 50 to 100 per cent due to plant competition. Some workers in the more humid re
gions, where moisture is often sufficient throughout the season for ordinary stands,
consider competition effects unimportant. For example, Stringfield (1927) found only
occasional disturbances in Ohio, while Garter and Odland (1925) failed to find evi
dences of competition in adjacent soybean rows on the West Virginia Station. Love
and Craig (1958) concluded that the effect of competition is not serious enough to
influence the yields of wheat and oats under New York conditions.
The influence of plant competition depends upen the test being conducted, but the
possible error from this source should be kept in mind constantly. It is a safer
procedure to eliminate or provide for this source of error than to be led to erron
eous conclusions by overlooking it.
A  I ntra plot Competition
VI. Uneven Plant Distribution in Plots
Plants within a plot are in competition with each other when some factor such as
moisture is present in insufficient quantities. Uneven plant distribution, with a
normal number of plants per plot, was studied in corn by Eiesselbach and Weihing
(1955) to determine whether or not this condition would alter acre yields. Corn was
planted in hills 3*5 feet apart so as to average three plants per hill. The three
systematically uneven distributions were planted so as to have 2*J, 135* and
123 J+5 plants in alternate hills. Essentially uniform stands of three plants per
hill were grown for comparison. During a ikje&r period the systematically variable
stands of 2k, 135, and 12 3 45 plants per hill averaged 50.6, 1*9,3, and 50.0
bushels per acre, respectively. The three variable stands averaged 50. bushels
while the uniform 3 plant stand averaged ^9*9 bushels per acre. In another trial,
these invest i gators tested a random variable stand by planting corn that germinated
100, 75; 60, and 50 per cents at adjusted rates to average three viable kernels per
hill. The plot yields for a single season were 2>+.96, 2p.50, 25.3^, and 25.12
bushels per acre for the respective germination per cents. From these data, it was
concluded that systematically and randomly variable stands did not affect the yields
so long as the same number of plants occurred on a plot. The authors caution that
"experience has indicated that stand irregularities materially greater than those
herein considered, such as are sometimes caused by rodents, worms, birds, and soil
washing, would undoubtedly increase plant variability and lower the yield."
A similar type of study was conducted by Smith (1937) on 2 Australian wheat fields
planted by a farm drill in which short lengths of drill row were harvested separate
ly. Variability of plant density as found in a drillsown field did not b;y itself
cause a decrease in yield of grain as compared to even spacing of seed. The correla
tion of yield and plant number per foot of drill row, which is invariably observed
3.58
in small grain fields, was said, to "be due to the effects of competition "between near
ly densities. He makes this statement: "The true correlation "between yield and
plant density per area may he positive , negative , or zero according to circumstances .
The yield from variable seeding may he less than, equal to, or even slightly greater
than the yield from even seeding, according to how near the even seeding may he
optimum for the given conditions and. how far the variable seeding may fall within or
overlap the optimum range within which plant density is of little importance."
VII . Differences in Stan d in Plots
The ideal condition for yield trials is a perfect stand, on all plots, hut this is
not always attained. Some allowance must he made for the lost area where the stand,
is injured, by outside influences, particularly with some crops. When the loss in
stand is due to the treatment, I.e., it injures germination, destroys part of the
plants, or in any manner is directly responsible for stand, the use of a perfect 
stand basis for yield calculations eliminates the effect of the treatment. The
lethal effect of the treatment may be a, definite part of the results obtainable and.
should, be given consideration. The effect of stand within plots is more of a prob
lem in crops like corn, sorghums, certain legumes, and sugar beets where the plants
are large, variable inter se", and subject to the influences of plant competition.
Less difficulty is experienced in small grains because the plants tend to tiller and
fully utilize the extra, space.
(a) Competition between Unlike Kills in Corn
Relative yields of one, two, and three plant corn hills uniformly surrounded
~bj threeplant hills was studied ~bj Xiessolbach (1918) (l'f'2'd). He found the yields
to be 61
82, and 100
per cents for the
two, and three plants per hill
It
obvious that the fewer plants per hill made some use of the additional space. In
another test, the relative yields of threeplant hills were compared when adjacent to
hills with various numbers of plants.
3plant hills surrounded by 3plant
hills except as Indicated below:
Average Grain Yield per Hill
Actual (lbs.) Relative (pet.)
Surrounded, by 3 plant hills
Adjacent to one hill with 2 plants
Adjacent to one hill with 3. plant
Adjacent to one blank hill
x . u ( p
1.098
1 ...pi
1 :22k
100
102
10?
nif
it is obvious that 3plant hills adjacent to blank or 1 or 2 plant hills tend to
yield higher than when surrounded by 3pl a tdc hills. In comparisons of inbred lines
and F_ hybrids, Brewbaker and lamer (1951) found, that a rather large error may be
introduced in yields where hills have reduced stands or are adjacent to hills which
lack in stand, however, under a 3plant rate in com it is generally conceded that
10 to 15 per cent of the stand may be lost before the yield is measurably reduced or
the experimental accuracy affected in ordinary yield, trials.
(b ) C ompetitive vs . NonCompetitive Yields in Su gar 3eets
Sugar beet tonnages are usually reported, as (I~) total weight of all beets on
a unit area, or as (2) a calculated yield, from "normally competitive" beets., The
beets which serve as the basis for calculation are those grown surrounded by neigh
bcrs on all sides at appropriate distances for the conditions imposed in the experi
ment., A study, of the response of sugar beets to increased space allotment was made
by Brewbaker and Deming 095. : >)« Their plants were grown in 20 inch rows and thinnt d
to 12 inches between plants in the row. Beets around a single blank space were foun^
to increase in weight sufficient to compensate for pfc>.2 per cent of the loss of a
single beet. They obtained increases of 28.7, 392, and. 950 per cents for beets
159
adjacent to one "blank space in the same row, • "between two "blank spaces in the same
row, and with "blanks on four sides, respectively. It is evident that the "beet weight
was greatly influenced "by the relative area available for its development. The re
gression of weight of "beets upon stand was essentially linear for stands between 25
and 75 per cent . For each 10 per cent increase in stand there was an increase from
O.76 to 2.10 tons beets per acre for the regression within blocks. There may be sit
uations where yields based on competitive beets would be in error, particularly in
poor stands and in spacing tests . Such instances have beer pointed out by Nuckols
(1956). He harvested actual and competitive beets on 25<+ plots where the stand
varied from 50 to 100 per cent . A greater difference in competitive and actual
yields was obtained for poor stands than for good stands. In fact, the mathematical
possibilities showed that there are only 35 P^r cent of competitive beets in a 90
per cent stand, "'0 per cent in an 80 per cent stand, and 5 per cent in a 70 per cent
stand. This indicates the greater possibility for error when competitive beets are
taken from poor stands. Nuckols also found an indication that there is a greater
difference between competitive and actual yields where the beets are closely spaced
than where more widely spaced in the row. It is obvious that in rate of spacing
tests, the method of selection of competitive beets is not the same for all plants.
(c) Stand Effects in Other Crops
In potatoes, Livermore (1927) reports that the yield of the two hills adja
cent to a blank may be kO per cent more than that for hills surrounded by hills.
Werner and Kiesselbach (1929) found 62.5 per cent of the loss in yield was recovered
in potatoes adjacent to onehill blanks. In alfalfa nursery plots, Tysial and Kies
selbach (1959) found for variable seed rates that stands tended to equalize after
k years. Considerable latitude in the amount of seed sown per row was possible with
out serious effects on comparative varietal performance.
VIII. Corrections for Uneven Stands
A great deal of attention has been given to possible corrections for loss of stand.
It should be emphasized that ,there is no entirely satisfactory method to correct for
uneven stands, it being better practice to prevent them so far as possible. One
method to avoid poor stands is to plant thick and thin the young plants to the de
sired rate. For example, where a stand of 3 plants per hill is desired in com, the
experimenter may plant 6 kernels per hill and subsequently thin the seedling plants
to 3 per hill. Most empirical methods for the correction of yields on a stand basis
are based upon plants surrounded by the normal stand, i.e., competitive plants.
Stewart (1919) (1921) gives a formula for the correction of stand errors in potatoes
where the stand is relatively satisfactory. The practice in corn experiments is to
harvest the entire plot without stand corrections when the stand is 90 per cent of
the theoretical or better. For less than that, it is usually harvested on a perfect 
stand basis. Kiesselbach (1918) (1923) selects only perfectstand hills surrounded
by hills with the same stand and computes the yields from these. Bryan (1933) found
that 26 per cent fewer hills were required to obtain any given degree of precision
with only perfectstand hills than with all hills regardless of stand. Adjustment
of the yields of perfect stand hills further reduced the number of hills required for
any degree of precision by 18.9 per cent. The procedure in the U. S. Department of
Agriculture for the uniform com hybrid tests is to adjust yields for missing hills
but not for minor variations in stand.
Probably the most satisfactory method for the adjustment of yields on the basis of
stand is by covariance in which the regression coefficients are calculated. Mahoney
and Baten (1939) have outlined its use for this purpose. When there is a fairly high
variation due to soil heterogeneity and no appreciable differences in stand, usually
nothing is gained by adjustment*
160
B  Interplot or Border Effect Competition
IX. Types of Inter^plot Competition
Many studies have been conducted to determine the border effects of adjacent plots.
The. committee on the standards for the conduct of field experiments for the American
Society of Agronomy (3935) makes this statement: "In a majority of soil experiments
and in many cultural and variety tests,, plot yields may be modified by contiguity to
other treatments, crops, or interspaces. Border competition in adjacent unlike plots
often raises some yields and lovers others." A vigorous variety may benefit when
grown next to a poor one, particularly in singlerow plots. The same type of error
may be introduced in rate and date of planting tests. As a result, multiplerow plots
are often used in experimental work with the border rows discarded. This procedure
is justified on the basis of experimental data which indicate that the yield order
may be changed when border rows are included in the plot yields, according to Arny
(1921) . In some fertilizer and cultural experiments alleys between plots are neces
sary because the treatment may spread to the next plot through faulty application.
X. Effect in Variety Tests
Most tests to determine the amount of interplot competition have been on the basis
of singlerow vs. multiple row plots with the borders discarded.
(a) Small Grains
It is concluded by Hayes and Amy (1917) that there is considerable competi
tion between rod rows of small grains when grown onefoot apart. This led to the
adoption, of threerow plots for small grain variety tests at the Minnesota station.
Comparisons of threerow plot yields with the central rows showed that the latter
are as accurate for yield determinations as attained by the use of all three rows.
Kiesselbach (I9I8) found that competition caused Big Frame wheat to yield 10. 3 and
12.!+ per cent too high in 1913 an  191^> respectively, when grown in alternate rows
with Turkey. Burt oats yielded lo and 38 per cent too high for these years when
grown in alternate rows with Kherson. Stadler (1921) found competition in small
grains to be more extreme between different varieties than between different commer
cial strains of the same variety. As a result, it is almost the universal practice
to grow small grains in multiplerow plots and discard at least one border row from
each side, at harvest for small grain nursery plots. The use of singlerow or 3row
plots with all rows harvested appears possible under humid conditions where competi
tion appears to be slight. (See Love and Craig, 1938)
(b) Cora.
As early as 1909? Smith (1909) found onerow plots too narrow for fair tests
in corn when varieties of diverse characteristics were planted in adjacent rows.
A variety with short stalks was at a disadvantage when grown next to a taller one
becau.se of shading; or a variety with "strong foraging powers" may compere more suc
cessfully for moisture and plant food over a weaker or slower growing neighbor.
Kiesselbach (1922) (1923) found that where large and small varieties of corn were
grown in alternate rows, the smaller variety yielded 66 per cent as much as the
larger one, and only k'J per cent as much when both were planted in the same hill.
The smaller variety yielded 85 per cent as much when planted in alternate 5row plots
ana the three center rows harvested for yield. That the smaller variety was being
robbed of light, water, and nutrients was shown by the yields where each variety was
surrounded by its own kind.
"(c) Other Crops
Competition between soybean varieties was studied by Brown (1922) in Connecti
cut. Twentyfive single row check plots of a small and early soybean variety averaged
161
26.9 "bushels of seed per acre. When the check was adjacent to larger and later
varieties like Mammoth Yellow, the checks averaged only 17.1 bushels or 63.6 per
cent as much as the average of all checks. In potatoes, he concluded that yields
were not influenced "by competition between singlerow plots.
In alfalfa, soliddrilled plots with a 7inch row spacing has been shown to
be definitely subject to serious interplot varietal competition. The work of Tysdal
and Kiesselbach (1939) indicates that the effects could be overcome when the border
rows were discarded at harvest. When the alley space between plots was widened to
12 inches a significant interaction between varieties was also prevented. The rela
tive yields from single or multiplerow plots with either 18 or 2^inch row spacing
likewise exhibited no significant differential interaction.
Immer (193^) made a study of the effect of competition between adjacent rows
of different varieties of sugar beets, i.e., "Old Type" and "Extreme Pioneer".
These were grown in alternate singlerow plots and also in 4row plots with the bor
der rows removed for yield deteiminations. When grown in singlerow plots the "Old
Type" brand yielded 3«78 + O.kk tons more per acre than "Extreme Pioneer." In ^row
plots, with the central two rows alone being harvested, the increase of "Old Type"
over "Extreme Pioneer" was only I.78 * O.Jl tone per acre. The difference between
these two differences was 2.00 * 0.5^ tons, a value that is significant. Thus, "Old
Type," the higher yielding sort, profited at the expense of "Extreme Pioneer" when
these two brands were grown side by side in singlerow plots.
In cotton variety tests, Christidis (1937) found that competition may cause
a definite bias in the estimation of comparative yields of cotton varieties. Han
cock (1936) tested two cotton varieties with diverse growth characteristics. The
varieties were: Acala, a late tall variety, and Delfos, an early semi dwarf type.
The varieties were arranged in these combinations with the series alternated: DDDDAD
and AAAADA. He observed that Delfos with Acala on on3.y one side (DDA) showed very
small differences when compared with themselves between their own border rows (DUD).
For instance, DDD as an average for four years produced only 1 ,h per cent more seed
than DDA, while AAA produced ^.01 per cent less than AAD. Where two rows of the
same variety are planted, only one row would be affected by a different variety.
Since he found this effect to be small, tworow plots were advocated with both har
vested for yield. Such a procedure may be satisfactory under conditions of abundant
moisture, but would be questionable where habitat factors are severely limited.
XI. Rate and Date Tests
Under most environmental conditions competition will exist between plots in rates
and dates of planting tests. Hulbert (1931) presents data to show that border effect
on outside rows increases as the rate of seeding is increased. The border effect on
Bed Bobs wheat was lk'J .85 per cent when seeded at the rate of three pecks per acre,
175.^1 per cent for five pecks, and I73.OI for seven pecks. Kiesselbach (1918)
tested two rates of planting for Turkey wheat, a thin and a thick rate. The thin
rate yielded 68 per cent as much as the thick rate when grown in alternate single
row plots, and 90 per cent as much when grown in alternate fiverow plots. Competi
tion between alternate singlerow plots for two rates for Kherson oats caused the
thin rate to yield 20 per cent too low in 1913 and 3^.3 per cent too low in 191^.
Nebraska White Prize corn was planted in alternate rows so as to obtain two and four
plants per hill. Due to competition the thin rate yielded relatively 29. and 9.0
per cents too low in different years. Similar results would be expected in date of
planting tests. Klages (I928) found a marked degree of competition in spacing tests
with sorghums. Yields of rows with dense stands profited at the expense of the yields
of adjacent rows with thinner stands. The degree of competition was influenced by
environmental conditions.
162
XII. Border Effect
Plants that grow along the aides and ends of plots are often more thrifty and vigor
ous than those in the interior. This is particularly true when the plots are sur
rounded by alleys. Border effect is considered here to mean the effect of "blank
alleys on the "border rows. The amount and extent of this border effect is important
in comparative crop tests.
(a) Small G rains
Amy and Hayes (1918) and Amy (1921) (1922) studied (1) the distance alley
effect is operative within plots, (2) the increase in yield duo to alley effect, and
(3) the influence of additional alley space on variety response. They used small
grain plots composed of 16 drill rows six inches apart . The yields of the "border row
wero compared with those of the center rows. Amy (1921) gives some typical data:
at s Who at B ar 1. e y
Description Bu. Pet. Bu^. Pet. Bu. Pet.
Outside border rows 65.58 1999 30,56 153.6
Middle border rows ■ 58.53 IJOik 25.75 127.1
Inside border rows ^995 1^2 .3 22.23 111.7
Central rows 32. oO 100.0 19.90 100.0
W.93
.213.5
k2,7k
136.5
35.56
IkS.k
22 . 92
100.0
As an average for three years the yields of outside rows of oats, spring wheat, and
barley expressed In per cent based on the yields of the central rows is 1998 and
that for the middle rows I38.O when the plots were surrounded by l8~Inch cleanculti
vated alleys Border effect was relatively unimportant when extended to the third
drill row. Knowledge that border effect is not uniform precludes the use of any
percentage figures derived in one place to reduce yields secured in another location
to a bordereffect free basis. Arnj (1921) further showed that the rank of a variety
may be changed due to "border effect. In all cases, plot yields were higher than
where these rows were eliminated before harvest. Hulbert end Eemsburg (1927) found
it necessary to discard two border rows from each side of small grain plots to remove
the error in border effect in variety tests. Competition effects were noticeably
increased when the adjacent plots were seeded at different rates. Hulbert, et al.
(1931) obtained similar results. Robertson and Koonce (193*0 studied border effect
on Marquis wheat grown in plots irrigated at different stages in its relationship to
yield when different numbers of border rows were included. The yield increased as
the size of plot increased but the percentage increase was uniform for the three
different treatments employed. Comparable yields were the same for plots of 10 rows,
and for 10 plus 2, k, or 6 border rows.
(b) Other Crop s
In kafir and milo, Cole and Hallsted (1926) obtained marked increases in
yield from outside rows. The excess yield was roughly proportional to the increased
available soil area. Recently, Conrad (1930 has called attention to the fact that
sorghum plants next to uncropped. areas may use soil moisture six feet away laterally.
A definite use of nitrates was made four feet away laterally for both sorgo and corn.
The influence of border effect on total dry matter per plot was studied at the Cen
tral "Experimental Farm (Ottawa) by McRostrie and Hamilton (157). In all cases,
border plants of Western rye grass gave an increased yield due to the influence of
the twofoot pathway which surrounded the plots. The increase in yield differed with
the strain under test, and varied from 6 to ^k per cent. The rank of the strains was
materially changed due to the wide variation in border yields. When theoretical plots
1/72.6acre in size were used for red clover and alfalfa forage yields, Hollowell and.
Heusinkveld (1933) found a serious experimental error in yield when border rows wore
included in the harvested plot. Their plots were composed of 8, 12, and loinch
163
alleys. The Inclusion of border rows increased the yield from 2.1 per cent to 20.0
per cent for red clover and from 1.8 to lt.O per cent for alfalfa. Border effect was
greater on the first than on the second alfalfa crop, hut varied greatly from year to
year under Ohio conditions. Rainfall appeared to he directly correlated with border
effect. These investigators concluded that the discard of two border rows would
effectively eliminate border competition on plots of this size. Similar results wero
obtained by Tysdal and Kiesselbach (1939) when they compared dissimilar adjacent al
falfa, plots that differed as to spacing of rows or plants. A soliddrilled block
with 7 inch row spacing was separated by a 7 inch alley space from a space planted
block with rows 2^inches apart. The adjacent border rows were compared with their
respective types of interior rows. The soliddrilled rows gave an excess yield of
'Jk per cent because of reduced competition on one side, whereas the spaceplanted
row was depressed 63 per cent in yield because increased competition. It is evident
that great care must be exercised in taking yields from adjacent rows that are
affected with respect to row space or density of stand.
XIII. Control of Inter plot Competition
Interplot competition can be controlled by several methods. Hayes and Garber (1927),
Kiesselbach (1918) (1923) and others give these recommendations: (1) group varieties
with similar growth habits, dates of maturity, etc., together; (2) use of multiple
row plots; and (3) discard outside border rows and ends at time of harvest. . Alleys
are sometimes used in closelysown crops such as small grains and forage crops to
facilitate harvest and to reduce mixtures. In small plots the borders should be re
moved, but in large field plots it is generally satisfactory to harvest the entire
plot and to include the additional alley space in the plot area. Untreated inter
spaces of sufficient width to avoid serious soil translocation are recommended for
permanent soil fertility, rotation, and tillage experiments. These alleys can either
be cropped or left bare.
References
1. Arny, A. C. The Dry Matter Content of Field Cured and Green Forage. Jour. Am.
Soc. Agron., 8:358363. 1916.
2. Border Effect and Ways of Avoiding It. Jour. Amer. Soc. Agron.,
1^:266278. 1922.
3. , and Hayes, H. K. Experiments in Field Technic in Plot Tests. Jour.
Agr. Res., 15:251262. 1918.
h. Further Experiments in Field Technic in Plot Tests. Jour. Agr. Res..
21:^83^99. 1921.
5. Brewbaker, H. E., and Immer, F. R. Variations in Stand as Sources of Experimenta n
Error in Field Tests with Corn. Jour. Amer. Soc. Agron., 23 :k6^h&l . 1931.
6. , and Deming, G. ¥. Effect of Variations in Stand on Yield and
Quality of Sugar Beets Grown under Irrigation. Jour. Agr. Res., 50:195210.
1935.
7. Brown, B. A. Plot Competition with Potatoes . Jour. Amer. Soc. Agron., 1^:257
258. 1922.
8. Bryan, A. A. Factors Affecting Experimental Error in Field Plot Tests with Corn.
la. Agr. Exp. Sta. Bui. 163. 1933.
9 Christidis, B. G. Competition Between Cotton Varieties: A Reply Jour. Amer.
Soc. Agron., 29:703705. 1937
10. Clements, F. E., Weaver, J. E., and Hanson, H. C. Plant Competition: An Analysir
of Community Functions. Carnegie Institution of Washington. I929.
11. Cole, J. S., and Hallsted ; A. L. The Effect of Outside Rows on the Yields of
Plot a of Kafir and Milo at Hays, Kansas. Jour. Agr. Res., 32:9911002. 1926.
i6h
12. Coleman, D. A., and Boomer, E.G. The BrownDuval Moisture Tester and How to
Operate It.. Dept. Bui. 1375, U, S. D. A, 192?.
13. Conrad, J. P. Distribution of Residual Soil Moisture and Nitrates in Relation
to Border Effect of Corn and Sorgo. Jour. Amer. Soc. Agron., 29:367378. 1937.
lit. Cook, W. II., Hopkins, J. ¥., and Geddes, W. F. Rapid Determination of Moisture
in Grain. Can. Jour. Res., 11:26^239, and kOOkk'j . 193)1.
15. Farrell, F. D. Easing Alfalfa Yields on Green Weights . Jour. Am. Soc. Agron.,
6:h2ko. I91I+.
16. Garter, R. J., and Odland, T . E. Influence of Adjacent Rows of Soybeans on One
Another. Jour. Amer. Soc. Agron. , 18:605607. 1925 •
17. Granthara, A. E. The Effect of Rate of Seeding on Competition in Wheat Varieties .
Jour. Amer. Soc. Agron., 6,:12'i128. 191*1. .
16. Hancock, E.I. Row Competition and its Relation to Cotton Varieties of Unlike
Plant Growth. Jour. Amer. Soc. Agron., 28:9^8957'. I936.
19 • Hayes, E. K., and Amy, A. C. Experiments in Field Tochnic in Rod Row Tests.
Jour. Agr. Res., 11:399^19. 1917.
20. , and Garber, P.J. Breeding Crop Plants, pp. 7579 1927 .
21. Hollowell, E. A., and Heusinkveld, D. Border Effect Studies of Red Clover and
Alfalfa. Jour. Amer. Soc. Agron., 25*779789. 1933 •
22. Hulbert, H. W., and Remsberg, J. D. Influence of Border Rows in Variety Tests
of Small Grains. Jour. Amer. Soc. Agron., 19:585590. 1927.
23. , , et al. Border Effect in Variety Tests of Small Grains. Idaho
Agr. Exp. Sta. Tech. Bui. No. 9 1931.
2k. Immer, F. E, Varietal Competition as a Factor in Yield Trials with Sugar Beets.
Jour. Amer. Soc. Agron., 26:259261. 193^ .
25. Kiesselbach, T. A. Competition as a Source of Error in Comparative Corn Tests.
Jour. Amer. Soc. Agron., 15:199215. 1 923.
25. Corn Investigations. Nebr. Agr. Exp. Sta. Res. Bui. No. 20.
1922 .
27. _ , , and Woihing, R. M. Effect of Stand irregularities upon the
Acre Yield and Plant Variability of Corn. Jour. Agr. Res. ^7:399i+l6. 1933
28. , Studies Concerning the Elimination of Experimental Error in
Comparative Crop Tests, Nebr. Agr. Exp. Sta. Res. Bui. 13 • I918.
29. ICLages, K. H . Yields of Adjacent Rows of Sorghums in Variety and Spacing Tests.
Jour . Amer . So c . Agron . , 2 9 : 582  599 . 1928 .
30. Livermore, J. R. A Critical Study of Some of the Factors Concerned in Measuring
the Effect of Select' on in the Potato. Jour. Amer. Soc. Agron., 19:857896.
1927 .
31. Love, H. H., and Craig, V. T. Investigations in Plot Technic with Small Grains.
Cornell U. Memoir 2l4. 1938.
32. Mahoney, C. E, , and Baten, W. D. The Use of the .Analysis of Covariance and its
Limitation in the Adjustment of Yields based upon Stand Irregularities 1 Jour.
Agr. Res., 58:317320. 1939 .
33" McKee, E. Moisture as a Factor of Error in Determining Forage Yields. Jour.
Amer . Soc. Agron . , 6:113117. . 1 91U .
3 1 '. McRostrie, G. P., and Hamilton, R. I. The 'Accurate Determination in Pry Matter
in Forage Crops. Jour. Amer. Soc. Agron., 19:2^32fjl. 1927.
35 Nuckols, S. B. The Use of Competitive Yield Data from Sugar Beet Experiments.
Jour. Amer. Soc. Agron., 28:92U93^. 1936.
36. Robertson, D. W. , and Koonce, D. Border Effect in Irrigated Plots of Marquis
Wheat Receiving Water at Different Times. Jour. Agi . Res., J+8: 157166. 193** •
37'. Smith, H. Fairfield. The Variability of Plant Density in Fields of Wheat and its
effect on Yield. Counc. Sci. and Ind. Res. Bui. 109 (Australia). 193 7 
38. Stadler, L. J. Experiments in Field Plot Technic for the Preliminary Determina
tion of Comparative Yields in the Small Grains. Mo. Agr., Exp. Sta. Res. Bui.
No. k9. 1921.
"■'■ "
165
39* Standards for the Conduct and Interpretation of Field and Lysimeter Experiments.
Jour. Amer. Soc. Agron., 25:803828. 1933.
he. Stewart, F. C. Missing Hills in Potato Fields: Their Effect upon Yields. New
York State Agr. Exp, Sta. Bui. 1*59, pp. 1*569. 1919.
kl m , Further Studies on the Effect of Missing Hills in Potato Fields
on the Variation in the Yields of Potato Plants from Halves of the same Seed
Tuber. New York (Geneva) Agr. Exp. Sta. Bui. 1*89. 1921.
1*2. Stringfield, G. H. Intervarietal Competition among Small Grains. Jour. Amer.
Soc. Agron., 19:971983. 1927
1*3. Tysdal, H. M., and Kiesselbach, T. A. Alfalfa Nursery Technic. Jour. Am. Soc.
Agron., 31:8398. 1939.
hk. Vinall, H. N.., and McKee, Poland. Moisture Content and Shrinkage of Forage and
the Relation of these Factors to the Accuracy of Experimental Data. Dept. Bui.
353, U. S. D. A. 1916.
45. Werner, H. 0., and Kiesselbach, T. A. The Effects of Vacant Hills and Competi
tion upon the Yield of Potatoes in the Field. An. Proc. Potato Assn. America,
16:109120. 1929.
1*6. Wiebe, G. A. The Error in Grain Yield Attending Misspaced Wheat Nursery Rows
and the Extent of the Misspacing Effect. Jour. Amer. Soc. Agron., 29:713716.
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1*7. Wilkins, F. S., and Hyland, H. L. The Significance of Dry Matter Determinations
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193B.
Questions for Discussion
1 . Why should pure seed \ ?. used in variety tests?
2. How may differences Iin acclimatization introduce errors in crop tests? How can
they be avoided?
3. When or with what crops or under what conditions is plant individuality a factor
to be considered in planning experiments?
1*. When is moisture content of the crop a factor of importance? How may the error
be eliminated or corrected?
5. Could you secure comparable forage yields by taking green weights? Why?
6. What are the advantages of the vacuum oven over an ordinary oven for securing
moisturefree weights?
7. Compare rapid moisture determining devices for cereals.
8. What is meant by plant competition? Who have emphasized its importance?
9 Is competition universally present in experimental plots? Is it always objec
tionable? Explain.
10. What effect does severe competition have on plants?
11. How can you reconcile the fact that some workers claim plant competition is a
fruitful source of error in experimental work, while others contend it is
negligible?
12. Do stand irregularities in corn affect the yield so long as the same number of
plants per unit area is involved? Explain.
13. Why may a variable stand in a wheat field yield as much as an evenlyspaced
stand? Explain.
Ik, Why is intra plot competition in small grains unimportant from the practical
standpoint?
15. What is the general effect in corn hills surrounded by hills with different num
bers of plants? Why?
16. What is meant by "normally competitive" in calculation of sugar beet yields?
17. What is the effect of adjacent blank hills on the weights of individual beets?
18. Under what conditions may yields from "competitive" beets result in errors in
yield?
166 . ■
19. What recommendations would you make as to correcting for uneven stands?
20. What is the general practice for the prevention of errors due to uneven stands
in .corn and sorghums?
21. How are stand errors generally corrected in corn plots?
22. What is meant by "border competition?
23. How may errors be introduced in variety tests by use of singlerow plots?'
2k. How could you .possibly justify tworow plots in cotton variety tests with no
borders removed? Singlerow alfalfa plots?
25. How does competition introduce errors in rate and date tests?
26. Under what conditions may it be desirable to have blank alleys surrounding plot:
2V. What influences do border rows have on plot yields?
2o. Is it always necessary to remove borders for the determination of plot yields?
Why? '■•.':
29. What recommendations would you make for the control of inter plot competition?
Pro blems
1.. Explain how to arrange and conduct an' experiment with 10 varieties of corn so as
to control both intra and interplot competition.
2. The yield of fieldcured hay on a l/luacre plot is *K30 lbs. The shrinkage sample
taken at that time weighed 3.8 lbs. After 3 weeks it weighed 3 .k lbs. Calculate
the yield per acre of the plot on an airdry be sis.
3 The yields (marketable ears) and stands of 6 strains of sweet corn for k replica
tions were as fellows: Data from. Malionoy and Bat en.
Yie
Id and Stand for
Strain Number:
Item
1
2 3
4
5
Replication 1 .
Held (x)
56
31 21
23
30
■ 60
Stand (y)
77
68 61
83
70
3)4
Replication 2 .
Held (x)
6k
29 32
20
59
30
Stand (j)
30
76 72
38
39
92
R epl i cat 1 on 3 •
Yield (x)
36
30 2k
18
60
>+7
Stand (y)
7*
.83. 82
78
78
78
Replication k.
Yield (x)
36
32 2k
19
39
30
Stand (y)
•57
6l 73
78
32
38
Yield Totals
212
122 101
80
208
2U7
Stand Totals
d. ju
268 283
327
289
31+2
Calculate the regression of yield on stand.
CHAPTER XV
DESIGN OF SIMPLE FIELD EXPERIMENTS
I. Criticisms of Agronomic Experiments
There are about 2300 agronomic projects in force in the different state, experiment
stations, "besides those carried on by the U.S. Department of Agriculture, and those
in related fields. In fact, twothirds of all agricultural experimental projects in
this country are agonomic. They have increased in number "by 50 per cent since 1920.
Frequently, this experimental work is criticised "by farmers and others. The criticise
may or may not be justified. Agriculture is sometimes looked upon as a "practical"
field in which results are sought rather than knowledge concerning the phenomena of
life. At other times, there is a genuine shortcoming in experimentation. Allen
(1930) states that fully onehalf of the agronomic experimental projects consist of
tests and trials of different kinds. Very littie ingenuity is involved in many of
them. Variety and cultural experiments are popular while many genetic studies are
merely field selection. Soil fertility experiments are often shallow. In many cases,
old methods of experimentation are used while in others the experiments are carried
too long.
A  Easic Principles in Design
II . Outline of Experimental Tests
A review of literature on the subject should "be the first step in the plans for an
experiment. This should "be followed "by a detailed outline in order to crystallize
the ideas of the investigator on the subject. Recently, Fisher (1937) has shown that
design of an experiment is inseparable from the statistical analysis of the data.
Certain objectives must be kept in mind in all agricultural experiments. These may
be enumerated as follows: (1) The tests should furnish a basis for recommendations
to farmers; (2) They should furnish occular proof of the beneficial results attained;
and lastly, (3) They should supply information on the fundamental causes of the
phenomena which the results are expected to demonstrate.
Several factors need to be considered in the outline of an experiment. These are
well described by Allen (1930) : (1) It should be definite and limited in scope.
(2) The problem should be subjected to competent persons for criticisms and sugges
tions. (3) Previous work on the subject should be familiar so that the investigator
can start work where others left off. (h) Next, he should ascertain the data essen
tial to the problem and devise means to secure and analyze them. (5) Then it remains
to test their applicability or sufficiency to the problem. Sometimes it is found
that progress is dependent upon the advance in related sciences.
III. Principle of the Extremes
Results should be secured over a wide range on either side of the optimum. Staple 
don (1931) bas made this statement: "I believe in all field experiments of a re
search nature we should go at each end far beyond what is deemed by practical men
to be the economic limit." The situation may be illustrated in a rate of seeding
test for wheat whore the optimum rate is approximately 5 pecks per acre. The pre
liminary test should include rates at regular intervals from the very lowest to a
maximum well beyond the point where the optimum is expected to fall, e.g.,
I67
163
1 peck 5 P R cks 10 pecks
minimum optimum maximum
The size of interval in tests is determined "by the amount of land, facilities,
character of the problem, or available finances. An increase in the size of the in
terval is justifiable as one goes from the optimum to either a minimum or to a maxi
mum. In the final test, it may he advisable to throw out the extremes ami conduct a
precise test around the optimum rate.
IV. Simple vs. Complex Expe riments
Experiments may he classified into several kinds based on the number of factors
studied at the same time. The formal experiment is sometimes preceded by a prelimi
nary test.
(a) Preliminary Test s
All preliminary experiments are necessarily empirical in nature. They give
the investigator an opportunity to detect faulty technique, inadequate methods, etc.
The final experiment can be planned to eliminate many of the shortcomings observed in
the preliminary test. A survey is sometimes used, for a preliminary test. A further
use of the preliminary experiment is to reduce the error in subsequent tests. (See
Wishart and Sanders, 1935) •
(b) Simple Experiments
One thing is studied at a time in the simple experiment. All factors are
kept constant or uniform, so far as possible, except the one under investigation.
This is the classical method of experimentation, i.e., the essential conditions are
varied only one at a time. R. A. Fisher has recently pointed out that this approach
is inadequate for many research problems because the lavs of nature may be controlled
and influenced by several variables. In his book on i: The Design of Experiments !I ,
Fisher (1937) makes this statement: "We are usually Ignorant which, out of innumer
able possible factors, may prove ultimately bo be the most important, though we may
have strong presuppositions that some few of them are particularly worthy of study.
We have usually no knowledge that any one factor will exert its effects independently
of all others that can be varied, or that its effects are particularly simply related
to variations in these factors". The simple experiment is justified when the time,
material, or equipment are too limited to allow for attention on more than one narrow
aspect of the problem. As an illustration of this type, an experiment can be set up
to determine the best variety of sugar beets to grow. Another could be designed to
determine the best fertilizers to apply, while a third separate experiment could be
relegated to the best cultural practices. The simple experiment is the one most com
monly used by investigators. It is recommended to beginners because it is less in
volved .
( c ) Combination and Complex E xperiments
More than one variable is studied at a time in combination experiments.
Examples of some of the more simple experiments of this type are: (1) rate and date
of planting tests, (2) the relation between time of planting and date of maturity,
(3) depth and rate of planting, in relation to yield, (k) fertilizer tests, etc.
Recently, the Rothamsted workers have advocated the complex experiment in which two
or more treatments are studied in all possible combinations. Yates (1935) states
complex experimentation is due primarily to R. A. Fisher who first suggested it in
1926. It is extensively practiced at Rothamsted and to a Lesser extent elsewhere.
Fisher (1937) claims two advantages of the complex experiment (factorial arrangement)
over experiments that involve single factors, viz., greater efficien cy and .greater
compreh ensiveness . A further advantage is that a wider inductive basis for conclu
sions is available. As an example, a complex experiment could be set up to determine
169
the responses of several fertilizers and methods of land preparation.
V. Replication
As previously pointed out, soil heterogeneity is the principal source of error in
the field experiment . It can he overcome theoretically "by replication which tends
to diminish the experimental error as well as to provide for an estimate of the mag
nitude of such errors. Fisher (1931) gives a diagram to show these relationships:
Replication
II
Random Distribution
skill
Local Control
Validity of estimate
of error
Diminution
of error
(a) Relation to Soil Heterogeneity
The decrease in the standard error of the mean of one variety or treatment
is proportional to the square root of the number of replications. Some workers have
argued that increased replication results in more heterogeneity due to the occupation
of a larger land area with the result that a point will he reached "beyond which fur
ther replication will give no further increase in accuracy. Fisher (1931) points out
that the experimental error is due only to the Irregularities within blocks and that
this difficulty is not effective when different treatments are compared locally with
in relatively small pieces of land. The number of blocks or replicates makes no
difference because the block effect may he removed by the experimental arrangement
(e.g. randomized blocks and Latin squares). Large "blocks presents a problem in it
self. The situation of large blocks led Hayes (I923) to make the statement that,
when a large number of strains are "being tested, it is necessary to use a large num
ber of replications to attain the same degree of accuracy as when a smaller number of
strains are "being compared. Special designs are advisahle for tests of a large num
ber of varieties or treatments.
(b) Duration of Tests
Replication in time is a necessary consideration in experimental tests.
Comparative results from various treatments or varieties are frequently modified or
even reversed in different seasons in response to climatic and soil variations and
to the prevalence of plant diseases, insects, and other pests. The American Society
of Agronomy (1933) recommends the continuation of a field experiment over a number of
years so as to give a random sample of such seasonal effects. As an illustration,
seasonal variahility at the Hays (Kansas) substation is greater than that due to
soil.
Crop
Variable Factor
Acre Yield (Bu.)
"Wheat
Wheat
Season
Soil
18J+
170
I p p
The standard deviation due to soil end season would tie: s 1 yl2~ + V" . T ae re_
duction in seasonal variation would require a replication of the test over a greater
number of years. Ordinarily , a minimum of 3 years should be required in a field ex
periment where a seasonal influence is important.
VI . P 1 ot Arrangem ent s
Each variety or treatment may he arranged either (i) in the same order in each repli
cate, or (2) entirely at random in each replicate. The former is called a systematic
distribution while the latter is designated as a random arrangement. Until rather
recently , systematic distributions have been generally used in field experiments.
Random arrangements have been advocated by Fisher (19:31) (1937) and the Eothamsted
workers who claim that randomization is necessary for a valid estimate of error. Re
gardless of the arrangement used; the various plots of a variety or treatment should
be arranged so as to adequately sample the experimental area. This usually leads to
certain restrictions on the arrangement.
(a) Random Ar rangement s
To justify random arrangements , Fisher (1931) states that uniformity trials
have quite generally established the fact that soil fertility cannot be regarded as
distributed at random but to seme extent systematically. As an average, nearby plots
are known to be more alike than those farther apart. Moreover , soil fertility dis
tribution is seldom or never so systematic that it could be represented ~bj a single
mathematical formula. As to the estimate of error, Goulden (1931) explains that it
depends upon differences in plots treated alike. Such an estimate will be valid only
when pairs of plots treated alike are not nearer together or farther apart than pairs
of plots treated differently. The total variance is made up of differences between
plots in both directions. When the differences between plots treated differently are
reduced by any sort of systematic arrangement one must automatically increase the
differences between plots treated alike, and vice versa, e.g.
V (total variance) ~ A (plots treated alike) + B (plots treated differently)
An alteration in either A or B will result in a similar alteration in the opposite
direction. Systematic arrangements which attempt to distribute the plots of any one
variety or treatment as widely as possible over the experimental area tend to reduce
B and increase A. Thus, the real differences between varieties or treatments are
reduced and the experimental error increased* An example of a random arrangement for
6 "varieties" in h replicates is as follows:
Replicate I: 572 lf86 31
Replicate II: 135628.7 k
Replicate III: l_C23p76k
Replicate TV: 7 _k 218 356
In practice, a set of random numbers such as those compiled by Tippett (1927) is use
ful to effect randomization of treatments or varieties. One may draw numbered, chips
at random or shuffle cards to obtain a random arrangement.
(b) Systemat ic Arra ngem ents
A systematic arrangement is the repetition of the varieties in the. same
order in each replicate. Correlation between adjacent varieties is likely under such
arrangements. However, systematic arrangements may be more practical in some experi
ments. Certain advantages have been given "oj the advocates of systematic arrange
ments: (1) Simplicity. It facilitates planting, harvesting, and notetaking opera •
tions. (2) It provides adequate sampling of the soil, i.e., allows for "intelligent
placement" of the various varieties or treatments, (3) Varieties may be arranged in
171
the order of maturity so as to facilitate machine harvest of field plots, (k) It may
he desirahle to alternate dissimilar varieties ("bearded and "beardless) so that mechan
ical mixtures can he detected in subsequent years. Systematic arrangement may he
effective in such cases. Thru the use of plots which provide for the elimination of
plant competition effects, systematic distribution loses one of its most serious
sources of systematic error.
The plot scatter on the experimental area is a matter of simple repetition
when the plots are all planted in a single series, viz.,
Replicate I Replicate II Replicate III
ABCLEFGH ABCDEFGH ABCDEFGH
As a rule, all plots cannot he placed exactly in one series, i.e., there are either
too few or too many. It is advisable to commence each block with a different variety,
especially when there is a soil gradient in the same direction as the series. This
eliminates the possibility that one variety will fall on the best soil in each block.
For compact blocks, the knight's move (one down and two over) is a common arrangement
to secure an adequate scatter, viz.,
Replicate Varieties
A B C D S F
G H
G H A B C I
I F
EFGEAB
D
I
II
III
(c) I nfluence of Arrangement on Error
Few data are available to show the relative accuracy of systematic and random
arrangements. The "Student" Fisher controversy in 1936 indicates that the problem
has not been fully settled. In a comparison of diagonal with random arrangements,
Tedin (1931) found that the degree of variability within 6 by 5 blocks was not in
fluenced by either arrangement in the estimate of error. However, he advised random
arrangements for the highest degree of scientific accuracy. In studies from uniform
ity trials with rice, Pan (1955) concluded that, with a systematic arrangement of
varieties, the deviations from mathematical expectation were too great to be explained
on the basis of random sampling. In a randomized arrangement, the number of differ
ences in yield between all possible comparisons of hypothetical varieties that fell
within a range of 0.5 cr, 1 .0 <r, etc., were computed. Satisfactory agreement with
mathematical expectation was obtained in two experiments, and poor agreement in one
(P = les3 than 0,01). On the other hand, Odland and Garber (1928) obtained somewhat
lower standard deviations from systematic arrangements than from the theoretical ran
dom arrangement. So far as small grains in nursery plots are concerned, Love and
Craig (1938) found the relative yields to be about the same for systematic and random
arrangements .
VII. Error Control
The differences between plots of a single treatment in a replicated experiment are
due partly to experimental error and partly to the average differences between repli
cates. The variability between replicates is irrelevant to the experimental test
when each variety or treatment occurs but once in a replicate. Therefore, the
variance due to replicates or blocks is generally removed from the error. The pre
cision of the experiment becomes greater when a large amount of the total variability
can be removed in this way.
is
172
The shape of plots and "blocks are also concerned in error control. Long narrow plots
are preferable within the "block so long; as the blocks themselves approach a square in
shape. The basic experimental designs are the randomized block and Latin square
arrangements. (See Goulden, 1939) •'
VIII. Randomized Blocks
The randomized block test is the simplest type of experiment where satisfactory error
control is obtained. This type of design is extremely flexible and can be used for
as many as 30 treatments. The principal restriction Ln this test is that the same
treatment should fail only once in each "block, the treatments or varieties being
arranged at random. The number of replicates or blocks depends somewhat upon the
number of treatments included in a block and the degree of precision desired. It if
preferable for 'the test area to be square in shape, altho this is not absolutely
necessary.
(a) Field arrang ement
A field arrangement for 10 varieties in t blocks could be as follows: \V
I 5 10 7 2 t 8 9 6 3 1
II 9 1823 10 3 7 6 t
III 6 1 2 9 8 3 ■ ■ 10 5 . 7 t
_IV 3 7_ _3_ _t _9_ 6_ _2 3_ _10_ _l_
For more than 30 varieties, special designs should be used. (See later chapters)..
(b) Computation of Sums of Sq uares
The yield data can be arranged conveniently for computation as in the table
below. (Data from Goulden, 199) •
Varieties
Blocks 1 2 5 t 5 6 7 8 9 10 Totals
I 34.0 16.0 jtvl it. 5 lo„5 29.9 28.6 16.0 17.3 23.I 232.0
II lt.0 11.0 20.5 13o 136 28,2 27.6 8.3 12.1 29.9 I8O.3
III 26.6 9.0 29.3 7.9 13. ^ 25.3 23.3 3.6 8.1 22,6 171.5
IV 18.5 H.9 21.0 13.2 8.9 28.8 Lo.3 95 10.317.7138.3
Totals 93.I H7.9 lOt. 9 31.1 56, t 112. t 96.2 39. t t7.8 92.9 7t2.1
Means 23. 12 11. 98 26.22 12. 78 it. 10 28.10 2t. 03 9.85 11.95 2322 18. 55
First, it is necessary to compute the sums of squares for octal., varieties, blocks
(or replicates) , and error. The correction factor is (Sx) /itf, or (?t2.l) /to =
550, 712. ti /to = 13,767.81.
Total = S(x 2 )  (Sx) 2 = l6 ; 279.27  13,767.81 = 2511. t6
N
Varieties = S(x v 2 )  (Sx) 2 = 62 /lit .0 1  13,767.81 * 1760.69
n N t
In this case, it is necessary to square the total for each variety and divide \rj the
number of values that make up each total to reduce the results to a singleplot basis.
v A set of random numbers such as table 6 in the appendix is useful to randomize the
varieties. In fact,, columns I, III, V, and VII were used.
173
Blocks ■S(ft b g )  (Sx) 2 = l4o,803.25  13,767.81 = 312. 51
m K 10
Error = Total  (varieties + blocks)
= 25U.W  (1760.69 + 312.51) = 438.26
The data are assembled in a convenient table as follows :
Variation Sums Mean
due to D.F. Squares Square s F value
• Blocks 3 312.51 104.17 6.1+2**
Varieties 9 1760.69 195.63 12.05**
Error 27 438.26 16.23 4.029
Total 39 . 2511.46
**Exceeds 1.0 per cent point, i.e., the vaxue of "F" which has a probability of 0,01
of occurring due to chance.
F = larger variance = 195.63 = 12 . 05
smaller variance 16.23
By reference to the F table, it is observed that the obtained Fvalue exceeds the
1.0 per cent point in both cases.
The other computations are as follows:
Standard error of a single determination (s) = V16.23 = 4.029
Standard error of the mean for each variety (o^) = s//n = 4.029 /V~^~= 2.0143
Standard error of a difference (cr d ) = 05^/2" = (2 .0143) (1 .l4l4) = 2.8486
Level of significance for 5 pet. point = (cr d )(t) (for 27 d.f .)
= (2. 8486) (2. 052) = 58453
In this case, 2.052 times the standard error of the difference gives odds of 19:1.
This value can be obtained from the "t" table by Fisher (1934) where "t" is taken for
the degrees of freedom for error at the 5 per cent point .
(c) Ap plication to Mean Comparison s
Tests are sometimes found in which the value of z or F, for the comparison of
variances due to varieties and error, just fails to reach the 5 per cent level of
significance. This would indicate that the differences between variety means were of
doubtful significance. In spite of this, certain differences between variety means
can often be found which exceed twice the standard error. The use of twice the
standard error (which gives approximately edds of 19:1 as the degrees of freedom
approach 60) would indicate that certain differences might be sign?f icant . However,
in such cases the testimony of the "z" or "F" test should be accepted as correct.
Twice the standard error is net a sufficiently stringent test for the comparison of
the greatest yield difference found in a large set of possible differences. Student
(1927) and Tlppett (1937) have both pointed out that, when the highest and lowest
values are compared, the conventional use of twice the standard error to obtain odds
approximately equivalent to the 5 per cent level of significance is no longer valid.
For example, with 10 varieties in the test the difference between the highest and
17*1
lowest varieties would need to reach 3 .2 times the standard error to lie on the 5
per cent level of significance. On the other hand, when !! F" is determined signifi
cant, the practice of using twice the standard error of the difference of two means
as a criterion for significance may "be too stringent when the means under considera
tion are contiguous in an arrangement of the variety (or treatment) means in order
of magnitude.
IX . The Latin Squa re
The Latin square design is very efficient whore a small number of varieties or
treatments is heing tested; hut It "becomes unwieldy for more than 1.0. Two restric
tions are imposed on the treatments In this design, i.e., the same treatment can
occur only once in 'the same row or column. The treatments are arranged at random
within these restrictions. The limitation of the Latin square for a large number of
varieties is due to the requirement of the same number of replications as treatments,
It should he emphasized that the plots need not he square in shape. (See Fisher and
Wis hart, 1930). This design gives error control across the field In two directions,
which always takes care of soil gradients, The most generally used Latin squares
vary from k by k to 10 by 10. Some data from an irrigation study with sugar "beets
will be used as an illustration of the Latin square arrangement in the field as well
as for the statistical analysis.
(s.) Field Pl ot Arrangement
The field layout for the 3 irrigation treatments (A,B,C,B, and E) was as
follows ;
Columns
1
1
2
3
k
5
•
E
D
A
B
d.
C
E
B
A
B
Rows
3
A
C
B .
E
D
k
D
B
E
C .
A
of
the
B
A
C
B
Tf>
Analysis
Data
The data for the irrigation stud;/ are compiled below, followed by the static
t i cal analy s is .
Tons I
>ee1
;s Per
Acre
Row
Row
1
o
7
i
(•
5
Totals
1
18.
32
i'E)
19
.ko
( T »
20.
66
(A)
22
63
(B)
18
.65
(c)
99
.97
2
20.
68
(c)
Ik
• 29
(B)
18.
32
(B)
20.
02
(A)
20
.58 (B)
9'+
39
3
26
ok
(A)
IT
M
(c)
21.
06
(B)
18
91
(s)
20
.03
(B)
103
53
k
22
31
(B)
dd
• 93
(B)
17".
15
(E)
17
lk
(G)
20
.62
(A)
100
ko
5
2k
kk
(B)
20
.2 L ;
(A)
lo.
92
(c)
19
73
(B)
lk
.07
(E)
97
kl
Column
Totals
112.
19
,9k
•kl
96.
61
98
kd
93
95
1495
70
Treatment
A
B
G
B
I
Totals
107.
59
111
■Ik.
92.88
100 .
35
82.
?k
Means
22.
35
23
.52
20.11
18.
38
16.
59
175
Correction factor = (Sx) 2 /n = (495.70) 2 /25 = 9828.7596
Total = S(x 2 )  (Sx) 2 = 10,007.8598  9.328.7596 = 179.1202
S
Rows = S(x r 2 )  (Sx^ = 9,858.1604  9,828.7596 = 9.4208
n K
Columns = S(x c 2 )  (Sx) 2 = 9,8759164  9,828.7596 = 4?.1768
n N
Treatments = S(x fc 2 )  (Sx) 2 = 9,9554952  9,823.7596 = IO6.7556
n N
Error = Total  (Rows + columns + treatments)
= 179.1202  (9.4208 + 45.1768 + 106.7556) = 17.7670
The data are assembled to complete the analysis:
Variation
due to
D.F.
Sums
. Squares
Mean
Square
Standard
Error
F
Actual
Value
Jfo Point
Rows
Columns
Treatments
Error
4
4
4
12
9.4208
45.1768
106.7556
17.7670
2.5552
11.2942
26.6889
1.4806
1.2168
1.59
7.65
18.05
5.26
5.26
5.26
Total
F = larger
24
variance =
26
179.1202
.6889 = 18.05
smaller variance 1.48o6
Since the computed Fvalue is greater than that for the 5 V er cent point, significant
differences exist "between treatments.
The other constants may be computed as follows:
Standard error of the mean (cr^) = s = 1.2168 = 0.5440 tons.
Standard error of the difference (a d ) = o^^lT = 0.5440 /2~ = O.77
Level of significance (for 5 pet. point) = 2.179 ©a = ( 2 .179) (077) = 168 tons.
The data may "be arranged as follows in summary form:
Treatment Mean Yield (tons)
A 22.55
B 21.52
C 20.11
D 18.58
E 16.59
Standard Error of the Mean 0.544
Level of Significance (5$ point) 1.68
176
B  Rel ation of Type of Experi ment to Design
% m Variety a nd Similar Teats
The variety test Is probably the most common type of agronomic field experiment.
Crop varieties are bested for yield in moat crop improvement programs to determine
which ones are superior under given soil and climatic conditions. Varieties are
known to differ as to the "best rate of planting. Less seed is required under dry lane
than under irrigated conditions due to the moisture factor. Car let on (1909) points
out that winter wheats tiller more than spring wheats and; when winter hardy, may be
sown, at a thinner rate. It is not always possible to overcome the objection of
differential response of varieties to different rates of seeding in a variety test.
It is usually safer to use a somewhat higher rate than that recommended to farmers
because variations due to unexpected causes will then have less effect.
Rate and date tests are sometimes combined wiith variety trials, or they may be con
ducted separately. The combined test permits a study of differential variety response
to different rates or dates. A rather wide range of rates on either side of the
optimum is suggested for rate of planting tests in order to determine the point of
maximum yield. A. test to determine the moat satisfactory dates for planting crops
is usually an exploratory stage in field experimentation to secure this information
for certain environmental conditions. Such tests are usually planted at a regular
mterval 01
tremi
fil^r
It
5 between dates from extremely early in the planting season to ex
)ue to occasional differential varietal response to time of ■planting
consideration should be given to the question of planting a variety series at several
different dates.
Ex
perimenti
the :
;;ned
with p to 10 varieties, while
varieties. For greater numbe;
should be investigated.
XI. Crop Sot at ion Experiments
as Latin squares for small precise tests
randomized blocks are commonly used l'or 10 to JO
s in a single experiment , incomplete block designs
esidual effects is
a study of r
to '^row all crops used in the rotation each year in order to
In crop rotations;, or other experiments in wh
made, it is neeessaip
obtain reliable results. Carieton (19Q9) early called attention to the fact that
this simple but essential matter had been entirely overlooked in many of the older
experiments. For accuracy in a rotation aeries , every stage or crop mu
every condition. Each year there must be as many plot
in the rotation. For example, in
(2) red clover, (p) corn, and (h)
experience
are crops or stages
as the.ro
kjear rotation of (1) oats seeded to red clover,
there must be four plots. The plots must
5X133',
ox over, {3 ) corn, an
be at least in duplicate in order to allow for the removal of soul variability,
quale replication is the greatest need in crop rotation experiments. In another
block in the same test there may be a plot of each crop, in continuous culture, al
though this is not always necessary. The crop rotation test, must be concreted over
a period of years so that the crop yields will be definitely influenced by the dif
ferent rotation treatments. Such a test might be laid out as follows:
i+year Rotation
Replicate I
Replicate II
(a)
Red
Clover
(b)
Corn Barley
(c) (d)
Corn Oats
(e) (f)
Continuous
Culture
••; ye ar Rot at i on
Oats
it)
Corn
(0)
Barley Corn
(g) (c)
Barley Oats
(d) (a)
iiture
ar I ev
(s)
del Clev
(b)
177
To compare this 4year rotation with a 3year rotation, it would, he necessary to
wait 12 years. For a 7 and 5year rotation, the results could he compared at the
end. of 35 years, etc.
XII. Cultural Experiment s
Cultural experiments include such tests as fall vs. spring plowing, methods of seed
bed preparation, surface vs. furrow planting, etc. Field plots are generally neces
sary for experiments of this type "because of the use of farm machinery. Many dryland
experiments are concerned with cultural methods. The same procedures for variety
tests are generally satisfactory in tests of this kind.
XIII. Fertilizer Experiments
The most reliable information on the fertilizer needs of soils may he obtained from
the field, experiment. Nutrient solutions and sand cultures are used in special
studies. The early fertility experiments at Rothamsted were concerned primarily with
the fertilizer value of certain mineral fertilizers as shown by increased crop yields.
The present longtime fertilizer experiments are concerned more with comparisons of
similar fertilizers, effects on crop plants, and efficiency of fertilizer practices.
The earlier workers often tested one fertilizer at a time, but many present workers
are inclined to favor more comprehensive tests, i.e., the inclusion of several fer
tilizers at more than one level. Most investigators use crop yield as the major
criterion n£ fertilizer response.
(a) General Types of Fertilizer Tests
Fertilizer Tests may be conducted for several definite purposes. (I) Defi 
ciency of Fertilizer Elements in a Field : Results of such tests are applicable only
to the field tested or, at most, to soil of similar type with similar previous cul
tural treatment. It is strictly applicable for the test year, since the crop grown
may modify conditions for the next season. (2) Efficiency of single Fertilizer
Elements ; For this type of test it is desirable to have the fertilizer elements
tested in minimum. (See Giles, 191*0. Several rates of a standard fertilizer can be
compared with one or more rates of a fertilizer that carries the elements in a differ
ent form. Equal rates of each fertilizer can be compared also. (3) C omparat iv e
Methods of Application ; This type includes tests on depth of placement, time of
application, placed to side vs. with the seed, etc. (h) Optimum Fertilizer Bala nce:
This type is concerned with fertilizer balance for various crops. It involves many
complications when made in the field because it is difficult to control or even
measure the fertilizer balance in a field. In such a study it is necessary to esti
mate by chemical tests the amount of the fertilizer elements furnished by the soil as
well as the amount applied. Probably the most practical method to make such a study
is to vary each element separately over a wide range qj several rates of application.
The regression of yield on amount of the element available (amount in plant plus
amount applied) may then be calculated. A further complication would be to test all
possible combinations of several fertilizers at different levels. The triangle sys
tem suggested by Schreiner and Skinner (1918) may be useful for the computation of
all possible combinations of three fertilizer elements (say P2O5J NH3; an & KgO) at
several levels. This triangle system should not be used as a basis lor the field
layout as originally advocated. Such a test should be designed as a factorial ex
periment. (See Chapter 19) . (5) Long Time Effect of Fe rtilizers: Such tests with
various forms of fertilizers are concerned with the physical and chemical properties
of soils as well as soil productivity.
(b) Design of Fertilizer Experiments
Several basic principles should be considered in soil fertility experiments.
A soil profile to a depth of 3 feet is highly desirable for each series of plots.
L/O
Before soil treatment experiments are "begun, the American Society of Agronomy (1953)
re commends that "representative samples of the soil and subsoil should b ) carefully
taken for such analyses as may he desired for future reference". In the matter of
plot design the Society cautions that "the lateral translocation of soil or ferti
lizer beyond the plot interspaces of soil experiments should, be avoided .
Since the manner of fertilizer application may .affect yields materially, due consid
eration should be given to this problem".
For fertilizer tests wh.;re 2 or more fertilizers are applied, at 2 or mere
levels, the factorial design is suitable. The factorial experiment , explained by
Fisher (193*0 <s Yates (1933)* Summerby (1937) and others, involves all combinations
of the fertilizers and levels (or amounts) of application. The study of interactions
is an important consideration in such an experiment. For example, suppose a ferti
lizer test is to be conducted with nitrogen,, phosphorus, and potassium at two differ
ent rates each. The rates can be designated by subscripts so as to give the 8 possi
ble treatment variants as follows:
I P a EC , %? E 0J NqP^ K o P %, %?iK , %?,.%, h,?^ arid 1^%.
Siren an experiment can be planned for a randomized block test o:c for some form of
the Incomplete block test. Goulden (193*0 gives some suggestions or. the design of
mere complicated fertilizer experiments. Residual effects duo to past fertility
treatments is discussed by Forester (1937).
XIV , Pasture Expe riments
In experimental pasture work, the investigator may desire to: (I) determine the
amount of herbage produced on an area by different pasture grass mixtures, (2) to
find out the influence of fertilizers on pastures as to yield and. survival of the
palatable species, or (>) he may desire to measure the influence of different grazing
methods on yield and survival. Replication of treatments is vital in any case.
One of the important technique problems is the comparative results from grazing and
mechanical harvest of herbage. Stariedon (iQpl) states that the. animal is the master
factor in pasture studies. He tethered sheep on small plots and moved thorn twice a
day in the Aberwystwyth pasture researches. Certain advantages are claimed for his
tethering method: (l) replicated plots are possible; (2) the experimental sheep are
handled and. examined twice per day; (3) grazing will be uniform, ana (k) the animal
capacity is increased per unit area. Schuster (1929) recommends at least h replica
tions and 3 animals per plot in pasture investigations. The use of grazing method.s
permits the effects of trampling on the vegetation te be measured. Pasture plots
maj be harvested mechanically, i.e., clipped, with a mower or with shears.. Brown
(1929) advocates the us^ of grass shears for small cages, .the lawn mower for grass
less than 6 inches high, and a mowing machine for tailor herbage. Several studies
have compared grazing and mechanical harvest of pasture plots.. Brown (1929) found
that the herbage of grazed and mowed, plots varied markedly in time due to animal pre
ferences. Animals void a large proportion of the fertilizer elements consumed in
feeds, particularly nitrogen and phosphorus. Thus, mowed pastures may be low in fer
tilizer elements when compared with grazed pastures. A high correlation between
mowed and grazed yields was found when the mowed areas were changed te the previously
grazed, areas every two or three years. Continuously clipped cages have yielded less
than annually mowed cages. Robinson, et ai (1937) ^ r und. a progressive decrease in
the yields of clipped permanent quadrats in relation to grazed areas.
Sampling methods are often involved in the design of pasture experiments. See Chap
ter 16 for further details en this abase, as well as the report of v'inall and others
(193V).
179
C  Incomplete Experimental Recor ded
XV. Missing Values in Experiments
In general, replicated field experiments are so arranged that the mean yield for all
plots that receive a given treatment provides the "best estimate of the effects of
that treatment. Sometimes the yields of some plots are lost or prove unreliable with
the result that the orthogonality of the original design disappears. Since th.4 treat
ment, "block, etc., effects are computed from the total yield of all plots in a given
treatment, "block, etc., it is necessary to interpolate the yield of the missing plot
in order to use the ordinary analysis of variance.
Allan and Wishart (1930) were the first to provide formulae for the estimation of
the yield for a single missing plot in either randomized "block or Latin square tests.
They arrived at their formula by the procedure of fitting constants by least squares.
Yates (1933) used a simpler solution by minimizing the error variance obtained when
unknowns are substituted, for the missing yields. The two formulae give the same
results, but the one by Yates also provided a method appropriate for the estimation
of the yields of several missing values. His formula is used here.
XVI. Calculation of Single Missing Valu e
A single missing value can be calculated for either a randomized block or latin
square test.
(a) Randomized Block Te st
Some data are given on the effect of date of planting on the yields of sugar
beets in which a plot value is missing. The yields are in tons per acre.
Date Block Number
Planted 1 2 3 k 3 Total 3
Sarly
22.3
21.8
19.7
21.2
Medium
13.3
iQ.k
18.5
21.5
Liat e
17.2
17.2
179
(18.8)
Very Late
14.9
12.6
13.1
Ik.k
20.0 105.0
17 • 3 9^ •
16.7 (87.8) 69.0
12.1+ 67.U
Totals 72.7 70.0 69.2 (75.9) 66. k (35^.2) 335. 4
;
It is assumed that the yield of the lateplanted plot in block k is missing. The
sums for block k and. for late planting are given below or to the right of the appro
priate block or treatment to show that they are the sums of only the known plots .
The values in brackets are filled in later.
 The formula for the estimation of yield of this value in a randomized block test is
as follows:
x = mM + m'M' Tx  .*. r  r   (l)
(m  1) (_•  1)
where x = yield of missing plot,
m a number of treatments
m'= number of blocks _^
M = sum of known yield.s of treatment with missing plot
M'= sum of known yields of block with missing plot
T x = total yield of known plots.
vThis portion is taken entirely from an. outline prepared by Dr. F. R. Immer, U. of
Minnesota.
180
Ip the sugar beet test used as an example,
x B H &9Q) ± %(2h ±} ~ 335. ^ = 18.8
(i  Dl5  1)
The yield, x  18.8, is inserted in the table after which the block, treatment, and
general sum are corrected accordingly. These figures are in the brackets. The
analysis of variance will be computed in the usual way, except that the degrees of
freedom for error and total have been reduced one. The degrees of freedom must be
reduced by one for each plot value interpolated. The analysis of variance is as
follows:
Variation Degrees Sums Mean Standard F Value
due to Freedom Squares Square Error (s) Observed 5 pet. point
Blocks k 13.043 3.2608 3.71 3.36
Treatments 3 lU9 . 6l8 Ho. 8727 56.69 359
Error 11 9677 0.8797 0.9379
Total 18 I72.338
(b) Latin Sq ue re Te st
The formula to be used for the interpolation of a single value in a latin
square test is as follows:
x = m(M r + M c + M+.)  2 T x ( 2 )
' (m1) (m2)
Where x = mis sing plot yield;
M r , M c , Mj. = totals of known yield? of the row, column, and treatment
from which the plot is missing;
m  number of treatments (also equals number rows or columns);
Tv = total yield of ail known plots,
XVII . M ore than One Missi ng Value
A method of approximation may be used, for irore than one missing plot yield. Three
plots are missing in the randomized block trial given below:
Pate
Planted
i_
p
7.
h
3
Total
Early
Medium
Late
Very Lat
e
22.3
10.3
17,2
14.9
21.8
(18.6)
17.2
12 . 6
(21,2)
18,5
17.9
13.1
T
21.3
(18.7)
20 . D
17.3
16.7
12.4
(106.5)
( 9 ] +.2)
( 87.7)
67. 4
83.5
73.6
69.O
Totals 72.7 (7C2) (70.7) (738) 66.4 (355.8) 207
51.6 i+9.5 57.I
The plot yields given in brackets have been assumed to be missing. As it is possible
to interpolate the yield, of only one plot at a bime, one must assume yields for all
missing plots except the one to be interpolated. First, suppose the medium planting
111 block 2 is interpolated. Per the early plot in block 3 and the late plot in block
4, one must insert the mean yield of the known plots for those two dates, or 21.3 an l
17.2 t on s , r e s p e c t i v e 1 y .
181
The formula for interpolation, in which have been substituted the values for the
first approximation for the yield of the medium planting in "block 2, is as follows:
x = mM + m'M'Tx = M75<6) ± 'JL2L&1 '" 3358 = 18.7
(m  l)(m'l) (k  1) (5  1)
The same procedure can "be followed for the early planting in "block 3> except that the
guess of 21.3 used "before should "be removed. The interpolated value (18.7) is used
for the medium planting in block 2, and the guessed value (17.2) for the late plant
ing in "block h. The grand total is corrected accordingly. The value of x in this
case is 21. 3.
In like manner the yield of the late planting in "block k is interpolated. This is
found to "be 18.7.
Since it was necessary to estimate the yields of two plots in order to start the
interpolation process, the values obtained will be somewhat in error. Therefore, the
values are re interpolated, using the values obtained by the first interpolation for
all but the plot yield being calculated. This is repeated until no further changes
take place. The values obtained in this case were as follows:
' . Approximations
Treatment Block 1st 2nd 3rd
Medium 2 18.7 18.6 .3.8.6
Early 3 21. 3 21.2 21.2
Late k 18.7 18.7 .18.7
The interpolated values did not change after the second approximation.
The interpolated yields are inserted in the above table (as shown in brackets) after
which the correct treatment and block totals are determined. The analysis of var
iance is then computed as shown below:
Variat ion
due to
Degrees
Freedom
Suras
Squares
Mean
Square
Standard
Error (s)
F Value
Obtained % Point
Blocks
Treatments
Error
k
3
9
11.923
I6O.306
8.309
2.9808
53. ^353
0.9232
O.9608
3.23
57.88
3.63
3.86
Totals
16
180.538
Three degrees of freedom have been subtracted from error and from total because three
plot yields were interpolated.
XVIII. ' Tests of Significance
The error calculated from analyses of variance, in which one or more plot values have
been interpolated, is a valid estimate of experimental error wnen the degrees of
freedom have been reduced by one for each value interpolated. However, • the variance
due to treatments is not entirely without bias, being always higher than it should
be. The significance of the test is accentuated, but the correction Tor this condi
tion is quite trivial for cases in which only a single value is missing. The bias
is more pronounced where many plots are missing.
Tests of significance by means of the analysis of variance are generally all bhat are
required. For a single missing plot, the treatment mean with the estimated value of
the missing plot will have an error as follows for a randomized block test:
182
°xt = 1
m'
m
(m ■ l)(m'  1)
(3)
Where m' = number of blocks, m = number of treatments, and s^ = variance of a single
plot calculated from error. The variance of the treatment mean would he s</m' where
no plot was missing.
For a single missing plot in a latin square, the variance of the treatment mean with
the missinp: value would he as follows:
m
1 + m  o 2    (h)
(m  l)(m  2) J
For rnore then one missing plot, these formulae are strictly applicable only to com
parisons between means where one contains no missing plot. To find the variance of
the difference between two means, both of which contain missing values, is rather
difficult.
Ref eren ces
1. Allen, E. W. Initiating and Executing Agronomic Research. Jour. Am. Soc. Agron.,
22:3^13^8. 1930.
2. Allen, F. E., and Wishart, J. A Method of Estimating the Yield of a Missing Plot
in Field Experiments. Jour. Agr. Sci., 20:399^06. 1930.
3. Brown, B. A. Technic.in Pasture Research. Jour. Am. Soc. Agron., 29:U68U76.
1957.
k. Carleton, M. A. Limitation in Field Experiments . Soc. Prom. Agr. Sci.,
pp. 5561. 1909.
5. Fisher, R. A. The Technique of Field Experiments. Fothamsted Conf., 15:1113
1931.
6. Statistical Methods for Research Workers. Oliver and Boyd. 5th
Ed. pp. 199251. 193V.
?■• Design of Experiments, Oliver and Boyd. 2nd Ed. vo . 15100.
1937.
8. , The Half Drill Strip System of Agricultural Experiments. Nature,
158:1101. 1956.
9. _ j and Wishart, J. The Arrangement of Field Experiments and the
Statistical Reduction of the Results. Imp. Bur. Soil Sci., Tech. Comm. No. 10.
I.95O.
10. Forester, E. C. Design of Agronomic Experiments for Plots Differentiated in
Fertility by Past Treatments. la. Agr. Exp. Sta. Res. Bui. 22o. 1937.
11. Giles, P. L. On the Plans of Fertilizer Experiments. Jour. Am. Soc. Agron.,
12. Goulden, C. H. Modern Methods of Field Experimentation. Sci. Agr., 11:681701.
1931.
13 , Statistical Methods in Agronomic Research. Can. Seed Growers
Assn. 1929.
14. Methods of Statistical Analysis* John Wiley, pp. ^551? and
11+211+8. 1959.
15. Hayes, E. K. Controlling Experimental Error in Nursery Trials. Jour. Am. Soc.
Agron., 15:177192. I923.
16. Love, H . II. and Craig, W. T. Investigations in Plot Technic with Small Grains.
Cornell Memoir 214. 1958.
17. Odland, T. E., and Garber, R.J. Size of Plat and Number of Replications in
Field Experiments with Soybeans. Jour, Am. Soc. Agron., 20:95108* 1928.
18. Pan, C. Uniformity Trials with Rice. Jour. Am. Soc. Agron., 27:279285. 1933.
185
19. Paterson, D. D. Statistical Technique in Agricultural Research. McGrawHill.
pp. 156188. 1939.
20. Robinson, R. R., Pierre, W. H., and Ackerman, R. A. A Comparison of Grazing and
Clipping for Determining the Response of Permanent Pastures to Fertilization.
Jour. Am. Soc. Agron. 29:3^9359. 1937 .
21. Schreiner, 0., and Skinner, J. J. The Triangle System of Fertilizer Experiments.
Jour. Am. Soc. Agron., 10: 2252^6. 1918.
22. Schuster, G. L. Methods of Research in Pasture Investigations. Jour. Am. Soc.
Agron., 21:666673. 1929.
23. Standards for the Conduct and Interpretation of Field and Lysimeter Experiments.
Jour. Am. Soc. Agron., 25:803828. 1933 .
2k. Stapledon, R. G. The Technique of Grassland Experiments. Rothamsted Conf . 13,
pp. 2228. 1931.
25. Student. Errors of Routine Analysis. Biometrika, 5:351' 19 2 7.
26. The Half Drill Strip System of Agricultural Experiments. Nature, I38:
971972. 1936.
27. Summerly, R. The Use of the Analysis of Variance in Soil and Fertilizer Experi
ments with a Particular Reference to Interactions. Sci. Agr., 17:302311.
1937.
28. Tedin, 0. Influence of Systematic Arrangement upon the Estimate of Error in
Field Experiments. Jour. Agr. Sci., 21:191208. 1931.
29. Tippett, L. H. C. Tracts for Computers XV Random Sampling Numbers. Cambridge
U. Press. 1927.
30. Tippett, L. H. C. The Methods of Statistics.' Williams and Norgate. 2nd Ed.
pp. 125139. 1937.
31. Vinall, H. N., et al. Report of the Joint Committee on Pasture Research. Am.
Soc. Agron. (mimeographed) 193^ •
32. Wishart, J., and Sanders, H. G. Principles and Practice of Field Experimentation.
Emp. Cotton Growing Corp., pp. 6085. 1935.
33. Yates, F. The Analysis of Replicated Experiments when the Field Results are
Incomplete. Emp. Jour. Exp. Agr., 2: 1291^2. 1933
3k. Complex Experiments. Suppi. Jour. Roy. Stat. Soc, 2:1812^7. 1935.
Questions for Discussion
1. What criticisms have been made of agronomic experiments in general? Are they
justified?
2. What justification, is there for the antagonism sometimes found between scientific
theory and practical facts?
3. What are the principal objectives in agricultural experiments?
k. What factors should be considered in the outline of an experiment?
5. What is the principle of the extremes? Illustrate.
6. In laying out field experiments in which one variable is continuous, what prin
ciple or rule should be followed with respect to the extremes?
7. Distinguish between preliminary and permanent experiments.
8. What is a simple experiment? Its limitations? Advantages?
9. What are combination or complex experiments? .Are they desirable? Why?
10. What sources of variation or error, other than that due to soil or season, may
occur in field experiments?
11. How does a random arrangement differ from a systematic arrangement?
12. Is soil heterogeneity systematic or random? Explain.
13. Upon what is the estimate of error based? How influenced by a systematic plot
arrangement ?
Ik. What are the advantages usually given for systematic arrangement? Random
arrangement ?
184
20
21
22
15. What is meant "by the "knight's move"?
16. Discuss the relative efficiency of systematic and random plot arrangements.
17. What is a randomized block test? What restrictions are imposed? Its limitations?
18. What is the Latin square arrangement of plots? What is the primary objective in
this arrangement?
19. What conditions should be observed in planning variety tests? What is a check
variety?
What precautions are necessary in crop rotation tests?
What are the limitations in fertilizer tests?
In rotation and soil treatment tests what should be the treatment of the check?
23. What is the law of the minimum? Its application to fertilizer tests?
24. What is a factorial experiment? Give an example.
25. Discuss grazing vs . mechanical harvest of herbage,
26. Why is it necessary to calculate missing values for the analysis of variance to
apply? . .,
27. How are the degrees of freedom modified when a missing; value is computed?
Problem s
1. Different amounts of fertilizer were: applied to sugar beets by the Colorado Experi
ment Station in 193° (Data from D. W. Robertson) in a randomized block trial. The
yields in pounds of sugar per plot for various amounts of treble superphosphate
applied per acre were as follows:
Phosphate
Treatment
None
100 lbs.
200 lbs.
300 lbs.
Totals
Block
I
II
343
I85.
358
413
393
.435
427 '
468
III
208
483
463
487
Total
730
1256
1291
1382
1321
150"
10
4l
£<*;
466
(a) Compute the analysis of variance for a randomized block experiment
(b) Determine significance by use of the "F" test.
(c) Compare the average .yields
the no treatment and 200 lb. treatment by
means of the standard error.
A rate of planting test with sugar beets was conducted in 1931 by H. E. Brewbaker,
The rates used were: 15, 20, 23, and 30 lbs. per acre. The experiment was de
signed as a 4 by 4 Latin square, the data for which follow:
Tons beets per
(3) 16.73
(4) 17.74
(1) 17.52
(2) 18.21
■ ore
~liTT
(2) 17.2c
38
(3)
00
'I o
,13
•53
Column
Totals 70.20
70.26
Row totals
( ]
10.35
(2)
15.27
63.73
(3)
18.83
(1)
16.94
70 . 71
(2)
17.97
(*0
18.31
71.95
(1)
177 1 !
(3)
16.61
72.09
70.89
67.13 278.48
(a) Compute the analysis of variance.
(b) Obtain "F" for a comparison of error with rows, columns, and treatments.
(c) Test the significance of 'F' : .
(d) Continue the analysis and compute the standard error of the mean, standard
error of a difference, and level of significance in case it Is justified.
185
3 Design a crop rotation experiment to show the effects of a legume in a rotation.
The rotations are as follows: (a) fcarley (seeded to alfalfa), alfalfa, alfalfa,
corn, and sugar heets; and (t>) "barley, corn, and sugar beets.
k. Four varieties of wheat were grown in I93O in a randomized block trial in 5 blocks
The yield of one plot was lost, (a) Calculate the. yield of the missing plot,
'b) Complete the analysis of variance.
Block
Variety
Kanred
5k.h
in .7
52.1
56.1
61.0
Cheyenne
ko.7
1^6.5
599
537
Tenmarq.
61.7
51.7
^35
61.9
58.7
Hays No. 2
55.5
50.6
61.9
1*5.1
72. k
5. The same k varieties were grown in a randomized "block test in 1937. The records
on 2 plots were lost. Calculate the missing values and complete the analysis of
variance .
Block
Variety
5
Kanred
5^.6
53.7
68.0
55.2
58.5
62.1
Cheyenne
66.3
60.9
6^.8
67.6
__._
66.2
Tenmarq.
58.5
575
kk.i
65.6
52.9
51.6
Hays No. 2
57.3
60.5
62.2
58.8
5^3
CHAPTER XVI
QUADRAT AED OTHER SAMPLING METHODS
!• Sam pling ; in Agr onomic Work
There* are times when it is impractical to lise the whole plot or plant population to
obtain a numerical determination of some characteristic of the experimental material.
In such cases as tiller number, yield? percentage dry matter, nitrogen or sugar in
the crop, it is mere practical to sample only a proportion of the whole. To quote
Wishart and Sanders (1955) : "The object is to obtain as close an estimate as we can
of the measure, which would, be obtained accurately, within the limits of experimental
error, had the produce of the whole plot been counted, weighed, or analyzed." The
sample must be representative and taken in such a manner as to assure that end. It
is also necessary to take into account the further source of error due to the sampling
process. Yields determined by sampling procedure arc not determined as accurately as
when the entire plot is taken, but it is often advantageous to sacrifice some accur
acy to save labor,
II. Theory of Sampling
The sampling distributions so far considered have been based on the assumption of
independence. The simple theory of errors does not apply when the variation is heter
ogenous and the extent to which the sources of variation are represented is not left
to chance. It has been shown in a randomized block trial that the variance due to
error is an unbiased estimate of the error variance of the infinite population from
which the data under consideration are a sample. The other items in the mean square
column (blocks and varieties) are not unbiased estimates of the respective variances
of the population. In fact, they contain the variance due to error as the degrees
of freedom become indefinitely largo. For example;, the estimated variance due to
varieties, in the theory of large samples, is made up of the true variance due to
varieties plus the variance due to error. This becomes important in statistics of
estimation as shown by Tippett (1957), Immor (1932, 1956), and others.
Suppose some data on protein in relation to different rate of planting treatments in
corn for 1931 be used to Illustrate the computations;
Method
Kate
Prote:
.n Per cent per
Sample v/
Planted
Planted
B3
.ock I
Block 11
Block III
Variety
(i)
(2)
(1) (2)
(1)
(2)
Totals
Golden Glow
Hills
o
10.357'
10.4o8
10A25 10. 522
10 . 1 00
10.043
oj. . op_;
:l it
a
9525
9.422
9.228 9.3^2
9.667
9.543
11 ii
•s
8.995
3.903
9'. 325 9.211
v < . vd. .
9.479
35^15
Pride North
3
IO.363
10.351
9.713 9553
9.627
59.212
n ii
4
0.171
92^5
9.399 9.576
9.057
9.052
<".<=; \r\ e
:l 1
5
9 . loo
9 . 120
9.171 9.211
8.527
8.504
53.690
Golden Glow
Drills
12^
IO.072
ic. 038
10.528 io.4o8
IO.38O
10.438
61.714
■M it
9
9.750
9 •"'/?
9.696 9.559
9^33
9.4.51
57.474
ii ii
o
8.8^6
3.778
a. old 3.oo4
9.143
9.080
93.512
ii H
3
8.482
8 . 590
9Jl79 9.422
go 1 ^
9.002
55.02p
Pride Worth
1 O
JL.w
9.872
9.929
10.009 9.384
9.827
0.724
l 59.7^i '■
H H
9
9.325
9A6B
8.853 3.892
9.H4
0. 14°
o4.8oo
■I ,i
6
O 'vO"i
0. ;Oy
3.761
9.523 9.365
8.832
3.802
34.384
.1 ;l
3
8.64.1
8.767
9.260 9,428
3.455
8.510
33.067
Totals
■
.31.323 3
.31,459 1
.33.236 133.237
131.123
131.o5q
791.432
vprotein = N (nitrogen) x 3«7 ^ Plants per hill N^ Inches between plants in row
186
187
The analysis of variance is set forth for the experiment in which two protein deter
minations were made on the shelled corn per plot. For simplicity, treatments will
he considered without regard to variety or method of planting.
The calculations for the sums of squares are as follows, the two samples per plot
being added together for the plot determinations:
S(x) = 791^52 (Sx) 2 /N = 7,14.56.7216
S(x s ) 2  (Sx) 2 /N = 7,^80.8645  7,1+56.7216 = 24.11+27
S(x ) 2  (Sx) 2 /N = 14, 961.1+1+26/2  7,456.7216 = 25.9997
P
S(xJ 2  (Sx) 2 /n = 208,799.0186/28  7,456.7216 = 0.5362
S(x t ) 2  (S X ) 2 /N = 44,852.7957/6  7,456.721b = 18.7444
The summary for the analysis of variance is as follows:
Variation
due to
Degrees
Freedom
Sums
Squares
Mean
Square
Standard
Error
F Value
Blocks
Treatments
Error
2
15
26
0.5362 .
13.7444
4.8691
0.1931
1.4419
O.1875
0.1+528
1.05
7/7O**
Total for Plots
41
25.9997
Samples within
Plots
"~Ts
0.1430
0.0054
O.O583
Total samples
85
24.1427
In the simple case where one sample is drawn from each plot with the treatment repli
cated for m plots, the variance of a treatment mean is V^/m, where V 2 , the mean
variance between plots approaches a 2 , the true variance of an individual plot as m
approaches infinity. However, when n samples are drawn from each plot, the variance
of a treatment mean is Vp 2 /mn, where V p 2 /n estimates a 2 the true variance of an
individual plot plus the true variance of an individual plot moan or a s /n. This
follows because a plot mean is now subject to variation due to more than one sample.
It is evident that o a 2 is the true variance of an individual sample taken from a
plot. The relationship may be shown as follows:
Y c
cr
+ a
ran
m
s
mn
_ i
m
(a
n
(1)
It should also be noted that
* cr
(2)
It is clear that ex 2 can be estimated from the above formula because V 2 and v are
both obtainable from the analysis of variance.
In the present experiment,
V 2 = O.I875, V 2 = 0.0054, n = 2, and m = 5.
P s
Mil
+ JL (o 2
vl
+ c r^ ) , and 0.0054
n
cr
188
Therefore ,
0,1873 v 1 ( cr ^ + oXlp> ) or . 0919 _ __ . cr 2 .
6 5 2 ' ■ p
The standard errors for plot and sample means are then calculated.
V 0.091 9 = 0.303s .....> or,, and /:>.003 J ; = 0.0583 > ct s .
The ratio, cc/crg is estimated as 0. 3032/0,0383 = 320. This indicates that the
variation between plots greatly exceeds that within plots or between samples, being
5.20 times as great.
Ill, E c o nomy i n _ S amp 1 ing
It Is of considerable importance to analyze hew the precision of ar, experiment, as
measured inversely . by i/m (cr'f: + c /n) 1" affected by varying m, the actual plot
p s
replications in the field, ana n> the number of sample;:; drawn from a plot. The rnosx.
important inference to be drawn is that the precision is mainly controlled 'by m, the
number of plot replications. Increasing the number of samples taken from the differ
ent plots can only appreciable affect the precision when o§ is not relatively small
v j. i j. .to o
as compared with cr . In the orosent problem 0.0034, the estimated value of o~'~ is
"0 6 to ■ ■
small compared with 0.0919s the estimated value of o " : . Hence, It must be concluded
' • j o
that to make more than one analysis on a. sample from a plot was unwarranted by the
small gain that would result,
( a ) C oiiiput at i on of Kuaib e r of Sampl e §_ or Eeplicate a
Thw required variance of the mean of a treatment (K) would be:
K = 1 (cr 2 f 0 2 ) '   ■  (3)
— n '
n
For the data in this problem,
K = 1_ (0.0919 + 0.003^ ) *0.0312
"3 2 '"
The computation, for different values of m or n, will give the number of replications
and number of samples per plot that will be necessary to reduce the variance of the
mean to a given level, i.e., K  0.03X2. The X values for the estimation of the
variance for treatments are as follows when the number of analyses per sample are
varied for three replications:
Humber of Samples (n) ^ril :i lill! ! _ll ilJ r! ' z. a.
1 O.0318
2 0.0312
3 0.0310
k '0.0309
3 O.0307
These data indicate the negligible effect when 1, 2, 3; ^? or 5 protein analyses are
laade from shelled corn camples.
( "a ) Determination of M in imum E xp en s e
Technical difficulties often prevent plot replication beyond a certain degree
In such cases it Is frequently worthwhile to strengthen the precision of the experi
ment by drawing replicate samples from the different plots. The number that should
be drawn depends on several factors. These factors are: (l) The variation between
plots as measured by oi in relation to o~, and (2) the cost of growing a plot ar
139
compared with the cost of obtaining and analyzing replicate samples per plot. The
time factor instead of the cost factor, or the combination of the two, should bo con
sidered in many types of experiments.
It is proposed to investigate how these relative costs determine a balance between
plot replicates (m) and. sample replicates (n) in order that a stated precision for
an experiment may be obtained at a minimum expense. Let C represent the cost per
plot replicated, and c the cost per sample replicate in the conduct of an experiment.
For a given treatment the total cost of plot replications will be mC, while the total
cost of sample replications will be mnc. Hence, E^ the total expense per treatment,
is given by:
E
mC
mnc
w
A pertain criterion of precision to be obtained may be represented by K = l/m (cr^ +
cri/n), where K = the required variance of the mean for a treatment. In order to^re
duce (4) to an equation with one variable, it is found that m = l/K (ct 2
a value which is substituted.
a'
/n),
Then,
E = l/K (cr 2
a 2 /n)(C + nc)
To reduce the total cost to a minimum, differentiate E with respect to n, and set the
equation to equal zero, viz.,
dE
dn
K
(a 2 + o a 2 /n)c + (C + nc)(n" 2 o~ s 2 )
=
co~ 2 +
P
2
c o~ s
n
 C cr
n s
 COrS =
n
n 2 ca = Co" 2
n 2 = G o 2
C cr
or
n =
C cr
c a
Thus, the total cost will be a minimum when,
r£= C a 2
(?)
c cr
In this case, n, and hence m, are determined to afford a most economical design. It
is worthwhile to note that n is determined to be independent of K, the precision
desired.
In the present experiment, the values are substituted in the above equation (5)j> viz
n = 2, o 2 = 0.0919, and 0% = 0.003^. The ratio of costs will be:
C = 0.0919 x 2 2 = 108.12
c 0.003*+
From the standpoint of expense, the analysis of a duplicate sample from each plot
would have been justifiable to produce the most economical design only if the cost
per additional plot had been 108 times the cost per analysis.
190
IV. Sampling, Practices
The important practical consideration in campling is that sufficient units should
he taken to give a reasonably accurate representation of the whole i In sampling
processes a small representative amount of the material io analyzed. For field plots.
Wishart and Sanders (1955) advise that the samples should amount to") per cent of
the plot at the very least. It must be recognized that the use of the entire plot
is the most reliable where it is feasible, as sampling can afford only an estimate
of the plot yield. Yates (1935) found yl per cent loss of information as an average •
of several experiments where sampling technic was employed.
Some of the practices for different crop material are discussed be. low. To determine
sampling errors it is necessary to draw at least two independent samples .
(a) Quadrat M ethods
Some form of quadrat is generally used for sampling yield trials, or for the
detailed study of vegetation. It was early pointed out by McCall (191?) that, while
the harvest of the entire plot is most satisfactory in yield trials, it is attended
with difficulties that make it practically impossible for plots away from the main
station. A form of quadrat was suggested as a. solution. The quadrat may be linear
for a certain length of row, often being unite of one foot, one yard, or one rod In
length. The type of quadrat most frequently used in range and pasture experiments
is a square area, usually a square motor or yard. The different kinds of area quad
rats are described oj Weaver and Clements (1929) •
1 Small Grains : The rodrow unit is widely used by American investigators to har
vest small ' grain plots by sampling methods. The English workers prefer one foot or
onemeter lengths of drill row.
The use of the rodrow, to secure a yield estimate for the entiro plot, was studied
by Amy and Garbor (1919). They harvested 9, 5, ana k rodrow samples from l/lO
acre plots of wheat and oats, and subsequently the entire plot. They concluded that
increases over the mean yield of the checks of 15.70 per cent for triplicate 1/10
aere plots, 9 '+9 per cent for the nine rod rows, 12,73 P er cent for the five rod
rows, and 14.M! per cent for the four rod rows were (on the average) probably signi
ficant. Nine rods removed from l/lOaqre plots were concluded to give practically as
accurate yield determinations as for the harvest of the entire plot,. It was admitted
that the amount of labor required to remove nine rod rows was about the same as for
the harvest of the entire plot. In studies with wheat and barley, Clapham (1929) ob
tained a standard error of less than 6 per cent for the yield estimate whore 30 one
meterlongth drill rows were harvested from l/jOacre plots. The red row method was
compared with meter lengths with six sets of five contiguous meter lengths. In bar
ley, the standard error for 30 met or lengths was 5.99 P^r cent, while the standard
error calculated from sets (rod rows) was 7*20 per cent.
The use of the square yard, in addition to the rod rev, has been studied by various
workers. Kiesselbach (I.917) determined the yield on lk entire l/^Oacre plots com
pared to 20 areas 32x32 inches (quadrat areas). It was concluded that 20 systemati
cally distributed areas may be safely substituted for the yield of the entire plot.
Amy and Sfeinmetz (1919) concluded that four or five systematically distributed
square yard areas removed, from l/lOacre plots gave approximately the same error for
yield as harvesting the entire plot.
2. Pota toes and Other Hill Crops: Yields of potatoes at Eothamsted were analyzed
both by sampling and by harvesting the entire plot. There were ].80 plants en each
plot. Wishart and Clapham (1Q29) reasoned that the individual plant was the logical
unit rather than a 'metrical unit. The actual number of plants necessary to determine
191
the plot yield depended upon the uniformity of the crop and the plot size, hut rare
ly was less than 10. A onein10 sample was inadequate for plots l/90acre in size,
it being necessary to take every third or fourth plant on a plot that size to give an
error of four per cent. It was concluded that there was little to gain "by the use
of sampling methods on plots less than l/20acre in size. It was better to harvest
the entire plot for yield. A pattern method has been used to sample sugar beet plots
at harvest time, the individual beet being the unit. Wishart and Sanders (1935) make
this explanation: "Suppose that there are 200 beets in a plot, and that it is pro
posed to take two sampling anits of 10 beets each. Two numbers between 1 and 20
(inclusive) are drawnsay k and lo. The plot is then covered by walking along one
row, back on the next, and so on, and the ifth, 24th, kkth beet pulled for one
sampling unit, and the l6th, 56th, 56th beet for the other sampling unit, totals
only being recorded in each case." Small samples of 10 roots for sugar analysis were
criticized by Johnson (1929) . Five samples of 10 roots from l/^Oacre plots gave a
difference as high as 2.2 per cent sugar. He concluded the results were unreliable
within one per cent of sugar each way. For 50beet samples taken in groups of 10 he
reduced the estimate of significance to 0.53^ P er cent and 0.593 V er cent sugar in
two experiment s .
3. Pastures : The area quadrat has been widely used in pasture investigations, the
usual size being a square meter. For the determination of the abundance and frequency
of species in range pastures, Hanson (193^) concluded that a quadrat twometers in
size was desirable. Stewart and Hutchings (195^) have suggested the Point Observa
tionPlot method for vegetation surveys. These plots are 100 square feet in arca ;
being marked off Idj a circle 5. 6k feet in radius. This method is claimed to be more
rapid than the ordinary quadrat method. It is also suitable for statistical analysis.
The vertical point method has been advocated recently by Tinney et al. (1937) • Two
horizontal pipes are mounted on legs 1.2 inches high, with a linear row of 10 holes
spaced two inches apart through which needles Ik inches long are moved up and down.
The point is pushed down until it touches a plant or a bare spot. The number of time's
a species is hit per 100 readings (needles) is expressed directly in per cent. The
needle may hit a number of different species, e.g., bluegrass '32, timothy 30, redtop
12, red clover 3> snd bare space 8. A modification is the inclined point method.
Others who have published on quadrats for pastures are Brown (193 7 )* an<i Fobinson,
et al (1937).
(b) Sampling from Bulk Material
So far sampling has dealt with plots during growth and harvest. Sampling of
harvested produce, or bulk material, is another important form found in experimenta
tion. In laboratory determinations, a sample of the material is mixed, and one or
two subsamples taken. A good method is to mix the heap of material thoroughly, to
divide it into four quarters and to reject, for example, the N.S. and S .\>J . quarters
mixing the other two again. The process is repeated until the bulk is reduced to the
size required for a sample.
1. Protein Determinations : Duplicate samples were taken in 2*+l cars of wheat by
Coleman, et al (1926) to determine the accuracy of sampling for protein. The cars
were sampled twice in 5 different areas with a gram probe. The contents of both
samples were composited separately and each reduced to 73 grams in size. Over 96
per cent of the tests varied less than 0.25 per cent in protein. In a study of sam
ple size the error was found to be less than 0.10 per cent when the samples weighed
30 grains or more. The error was higher for smaller samples. Bartlett and Greenhill
(I936) and Leonard and Clark (1936) found that one protein determination per repli
cate reduced the error more rapidly when the number of replicates was increased than
by the use of duplicate laboratory analyses. The latter workers found the cost ratio
of plot replicate to sample replicate for protein determinations in corn. The analy
sis of a duplicate sample from each plot would have boon justified in producing the
192
most economical design only if the cost per plot had been 108 times the; cost per
analysis.
2. Shr inkage Samples in R e rage: Two or three samples per plot were found by Wilkins
and Inland (193^) to accurately measure the water content of forage on individual
plcts of alfalfa and red clover. Samples 2 to k "pound's in size were considered best.
3. Purity and Germinatio n Tests in Seeds : Tests for germination and purity in seed
analyses are affected "by personal, sample, and random ei~rcrs . Standard rules (I927)
for seed testing specify minimum sample s3z.es as follows: (1) Two ounces of grass
seeds; (2) Five ounces of red or crimson clover ; alfalfa, rye grasses, breme grasses,
mi.ll.et, flax, rape, or seeds of similar size; (3) one pound of cereal, vetches, or
seeds of similar size. Collins (1929) has set forth the procedure for statistical
anal3'ses of purity and germination tests .
References
1. Anonymous. Rules for Seed Testing, Dejpt . Cir. ko6 } U.S.I. A. 1937
2. Amy, A. C., and Garter, R. .J. Field Technic in Determining Yields of Plots of
Grain "by the Bod Row Method. Jour:, Am. Sue. Agron., 11:3347. 1919.
3. Amy, A. C, and Steimaetz, F. H. Field Technic in Determining Yields of Experi
mental Plots by the Square Yard Method. Jour. Am. Soc. Agron., 11:81106.
1919 .
k. Bartlett, M. S., and Greenhiil, A. V, The Relative Importance of Plot Yariat.u.n
.and of Field and Laboratory Sampling Errors in Small plot Pasture Productivity
Experiments. Jour. Agr. Sol., 238262. 193o.
p. Brown, B. A, Tochnic in Pasture Research. Jour. Am. Soc. Agron., 29:^68^76.
1937.
6. Clapham, A. P. The Estimation of Yield of Cereals by Sampling Methods. Jour.
Agr. Sci., 19:214233. I929,
7. Coleman, P. A., et al. Testing Wheat for Protein with a Recommended Method for
Making the Test, Bept. Bui. 1460. U.S.D.A. I926.
8. Collins, G. P. The Application of Statistical Methods to Seed Testing, Cir. 79
U.S.D.A. 1929.
9. Fisher, S. A. Statistical Methods for Research Workers, Oliver and Boyd.
(Vth edition). 1933 •
10. Hanson, H. C. Size of List Quadrat for Use in Determining Effects of Different
Systems of Grazing Upon Agropyron Smithii Mixed Prairie, Jour. Agr. Res.,
kl:3!+9360.
11. Immer, F. R. Sampling Technic. Mimeographed outline, U. of Minnesota. 193 c »
12. Immer, P.P. A Study of Sampling Technic with Sugar Beets. Jour. Agr. Res,,
^:633oV7. I.932,
13. Immer, F. P., and LeClerg, P. L. Errors of Routine Analysis for Percentage of
Sucrose and Apparent Purity Coefficient with Sugar Beets taken from Field Ex
periments. Jour. Agr. Pes., 92: oG^PIi 1936;
Ik. Johnson, S. T. A Pete on the Sampling of the Sugar Boot, Jour. Agr. Sc,
19:311315. 1929.
13. Kiessellach, T. A. Studies Concerning the Elimination of Experimental Error in
Comparative Crop Tests, Nebraska Research Bui. 13. 1917
16. Leonard, ¥. H.., and Clark, A. Protein Content of Corn as Influenced by Labora
tory Analyses and Field Replication. Colo. Exp. Sta. Tech. Bui. .19 « 1936.
17. McCall, A. G. A Pew Method of Harvest ine: Small Grain and Grass Plots. Jour. Am.
Soc. Agron., 9:138lik0. I9I7.
18. Robinson, 'P. P., Pierre, W. E., end Ackerman, P. A. A Comparison of Grazing .and
Clipping for' Determining the Response of P; remanent Pastures to Fertilization.
Jour. Am. Soc. Agron., 29: 319 339. 1937
195
19. Stewart, G., and Hutchings, S. S. The Point Observation Plot (SquareFoot 
Density) Method of Vegetation Survey. Jour. Am. Soc. Agron., 28:714722.
1936.
20. Tinney, F. W., Aamodt, 0. S., and Ahlgren, H. L. Preliminary Report of a Study
on Methods used in Botanical Analyses of Pasture Swards. Jour. Am. Soc. Agron.,
29:8353^0. 1937. .
21. Tippett, L. H. C. The Methods of Statistics. Williams and Norgate, pp. 125130,
and 20^226. 1937
22. Weaver, J. E., and Clements, F. E. Plant Ecology, pp. lOlj2. 1929.
23. Wilkins, F. S., and Hyland, H. L. The Significance and Technique of Dry Matter
Determinations in Yield Tests of Alfalfa and Red Clover. la. Agr. Exp. Sta.
Res. Bui. 2^0. I938.
2k. Wishart, J., and Clapham, A. P. A Study of Sampling Technic: The Effect of
Artificial Fertilizers on the Yield, of Potatoes, Jour. Agr. Sci., 19:6006l8.
1929.
25. Wishart, J., and Sanders, E.G. Principles and Practice of Yield Trials. Empire
Cotton Growing Corp., pp. k2k5, and 8597. 193&.
26. Yates, F. ; Some Examples of Biased Sampling. Ann. Eugenics, 6:202213. 1935.
27. Yates, F., and Zacopanay, I. The Estimation of the Efficiency of Sampling with
Special Reference to Sampling for yield in Cereal Experiments. Jour. Agr. Sci.,
25:5^5577. 1935.
Questions for Discussion
1. Under what conditions may it be desirable to take samples rather than use all the
available material?
2. Are yields determined from samples as accurate as when the entire plot is har
vested for yield? Why?
3. In a randomized block trial what makes up the variance due to varieties?
k. When can an increase in the number of samples effect an appreciable increase in
precision?
5. What were the conclusions in the shelled com data on the number of samples for
protein analysis?
6. What factors influence the number of samples that should be drawn so far as the
precision of the experiment is concerned?
7 Explain how to determine the most economical design from the standpoint of cost.
8. Under what conditions are quadrat methods used for small grain harvest?
How many quadrats are usually advised for l/lCacre plots?
9. Compare the rod, meterlength, and area quadrats for small grains.
10. What is the logical sampling unit for potatoes or sugar beets? How many samples
should be taken?
11. Describe how to use a pattern method of sampling for sugar beets and similar
crops .
12. Describe the different quadrat methods used in pasture studies.
13. Give sampling precautions and technic to use in bulk material.
\k. What errors may be introduced in seed testing? Give the sample sizes generally
used for analyses.
Problems
Four varieties of crested wheat grass were grown in a randomized block trial in plots
l/80acre in size. Four replicated plots of each variety were grown, the yield data
being obtained from six quadrats per plot. The yields in pounds per square yard sam
ple are given below. (Data from Dr. T. M. Stevenson, U. of Saskatchewan).
194
Yields of Crested Wheat Grass
Square
Yard
Block
Mecca:S,l
CW.G; :S 10 CW.G. :S11
CW.G. :Uns
(No.)
(No.)
(lbs.)
(lbs.) (lbs.)
(lbs.)
1
I
0.52
0.68 0.48
0.58
2
oJi9
0.62 0.55
O.58
3
0.59
0.70 0.46
0.61
4
O.36
. 70 . 58
O.63
P
G.28
0.62 0.51
O.65
6
G.49
0.66 0.38
O.71
I
IT
0.61
O.77 0.44
0.68
2
0.49
0.91 0.48
0.43
3
O.52
0.89 0.U9
0.75
4
0.56
0.95 0.61
O.71
5
0.57
. 77 . 58
0.65
6
0.49
0.77 0.4l
0.68
1
III
. 52
0.Q;: 0.27
0.42
o
0.42
O.77 0.61
. 51
5
0.66
0.46 0.44
. 58
4
0,57
o.3i 0.51
0.54
o
. 59
0.58 0.61
. 66
6
0.56
0.35 0.4l
0.58
T,
17
0.42
0.70 0.55
. JO
2
0.51
0.37 0.72
0.30
3
0.V7
O.53 0.63
0.44
4
. '50
0.60 0.65
0.66
ir
0.55
. 64 . 48
0.63
6
.26
' 0.33 O06
o.4i .
1. Calculate the analysis of variance for the crested wheat grass yields for a sub
division of i
within plots
division of the total variation into blocks , varieties, error , and square yards
2. Compare the variance due to samples and that due to replications. Make a state
ment on the number of samples and number of plot replicates that you would recom
mend in a subsequent experiment .
3. Determine the most economical design from the data at hand, i.e., the ratio of
replicate cost to sample cost.
uH
CHAPTER XVII
COMPLEX EXPERIMENTS ^/
I . Use of Complex Experiments
The present trend in field experiments is toward somewhat complicated designs which
permit the study of several factors in one largescale comprehensive experiment.
There are several advantages of the complex experiment. (1) One that includes sev
eral treatments in all possible combinations permits a broad basis for generalization
due to the fact that the interactions, as well as the main effects, can be studied.
It is obvious that field experimental results may be influenced materially by en
vironmental conditions with the result that a combination of factors may provide a
more satisfactory answer to the problems undor study. (2) The degrees of freedom
for error variance are higher than would be the case for single experiments designed
to study each factor separately. This leads to greater precision in the results.
(See Paterson, 1939).
The value of a complex experiment depends upon a careful analysis of the problem and
the various treatment combinations to be tested. The amount of complexity introduced
depends also upon the facilities and funds available. It is a safe precaution to
keyout the degrees of freedom for the various factors to be tested before field
work begins to be sure that the proposed plan is satisfactory. After the data are
collected the investigator should make certain that the data are sufficiently homo
genous to combine in a single test. (See paragraph VII).
H. Application to a Barley Variety Trial
In agronomic tests of cereal crop varieties it is often desirable to conduct the
trials at various points in the area under consideration and to carry them on for a
period of years. Some data collected by Immer and others (193*0 on the yield of
barley varieties tested in randomized blocks in h locations in Minnesota for a 2 year
period will be used to illustrate the method of computation. The data are based on
6 square yard samples harvested from each plot of approximately l/^Oacre each. Each
test consisted of 3 randomized blocks. The same 5 varieties were tested at Univer
sity Farm, Waseca, Crookston, and Grand Rapids for the years 1932 and 1935. The
yields in bushels per acre for each plot of each variety are given below in Table 1.
H* From Dr. F. R. Immer, University of Minnesota, with minor modifications.
195
196
Table 1. Yields of five varieties of barley, replicated 3 times in each of 4 loca
tions in 1932 and 1935.
Block
Number
Tot. for
I
IT
III
Tot.
I
II
III
Tot.
"both years
Univ. P
arm  1932
Univ.
Farm 
1935
Manchuria
197
31. 4
29.6
80 . 7
45.5
50.3
60.0
155.8
236 . 5
Glabron
28.6
38.3
•43.5
110. h
47.5
4.1.4
49.4
138.O
248,4
Velvet
20.3
27.5
32.6
80. k
54 . 2
52.3
64.5
171.0
251 .. 4
Wis. #38
27.9
4o.o
46.1
114 .
62.2
53.1
74 • 7
190.0
304.0
Peatland
22.3
30.8
31.1 ■
84.2
47.4.
57.8
50.5
155.7
239.9
Total
118.8 168.0 182.9 4697 256.3 254.6 299.1
810,3 1230.2
Manchuria
Glabron
Velvet
Wis. #38
Peatland.
40.8
44.4
44.6
39.8
71.5
Wase ca
29.4
34 . 9
41.4
3Q.2
47. <
1932,
30.2
35 • 9
26.2
29.1
554
Total
241.1 .192.5 174
100 , 4
113.2
112,2
103.1
1745
608.4
539
637
5". 9
74.2
51.1
Waseca  I.935
58.8 47.7
52,2
56 . 4
67.O
45.O
61 . 1
591
756
47.3
160 . 4
177.0
169.4
216.3
143 . 4
260.8
290.2
281.6
324.0
317,9
296.8 301. a 268.3
867.0 14754
Manchuria
Glabron
Velvet
Wis, #38
Peatland
Total
34,7
23.8
29.8
2.7.7
43.O
164.0
Crooks ton
1932
2o.l
28.7
38.4
27 ..6
7,0 '7
.■■ c  •_ I
156.5
35.1
2]. .
28.0
20 . 4
32.0
136.5
98.0
73.5
96.2
75.7
107. 7
457,0
rooks 1
:on  1
935
42.1
47.1
30.3
120
218.9
38.8
29 . 4
305
93
7
172.2
42.1
4 .
393
121
21.8.1
44,3
43.5
47 • 7
135
5
211.2
53.9
51 . 8
50 . 3
156
263.7
21.2
211 . 8
199.1
632
1
1089.1
Man churls
Glabron
Velvet"
Wis. #38
Peatland
Total
20.2
13.2
24,5
19.O
27.6
Grand, Ra.pids
~16~.<
30,2
20,5
41.6
18.4
30 .
9.6
30
Cr+  O
OO 7
1932
104.5 140.7 103.5
)
4
26
Grand
Rapids
£> •■''
26.5
327
43
3
21
h
: 18.7
24.1
96
7
1
20
7
26 . 8
30 . 4
62
20.
7
236
30.9
30
3
32
6
4o.O
34 . 2
.43
7
122.
135.6
152.3
 1 9"
85.8 152,
64.2
77 • 9
85*2
106.8
409.9
107.5
174.6
137.2
187.1
'58
p <
Total 4
Stations
628.4
6577
5977
1883.0
8Q6
903.9 913.8 2719.5 4603.3
III. Analysis of Tes
Into Corrroonents
The analysis of a complex experiment of this type is merely an extension of the
analysis of variance as applied to the randomized block test. The various factors
t of ether with their degree
of freedom may be represented as follows: It is noted
that all block x variety 1  interactions are included in error,
1
The symbol (x) in this connection denotes interaction,
197
Variation due to:
Degrees of Freedom
Slocks
Stations
Years
Variet
ies
Interact'
Lons of
Variet
iei
3 x Stations
it
x Years
(i
x Stations
x Years
Stations
x Years
Blocks
X
Stations
ii
X
Years
»i
X
Stations x Years
it
X
Varieties
ii
X
Varieties x
Stations
ii
X
Varieties x
Years
it
X
Varieties x
Stations x Years
)
) Srror
)
)
2
3
1
4
12
k
12
3
6
2
6
8 )
24 )
8 )
24 )
64
Total
119
There will be a total of 119 degrees of freedom for the combined test since there are
120 plots. The degrees of freedom for the main effects will be one less than the
number of blocks , stations, years , and varieties , respectively. The degrees of free
dom for interaction is obtained by the multiplication of the degrees of freedom for
the variables involved. For example, varieties x stations will be (4) (3) = 12. For
the second order interaction, varieties x stations x years, the degrees of freedom
will be (4)(3)(l) = 12. After the degrees of freedom are keyedout, the remainder
of the computation must be made in accordance with this plan.
IV. Computation of the Sums of Squares
The correction factor, (Sx) 2 /N, is computed for the total yield of the 120 plots, i.e
(1+603. 3 2 /l20 = 176,586.4241. This factor will be used for the entire test.
For the total sum of squares, the 120 individual plot yields are squared and summed.
This value (Sx 2 ) is equal to 200, 87935. The correction factor is then subtracted,
viz., 200,879.35  176,586.4241 = 24,292.9259 XIII. ^
The data for the remainder of the computations are grouped from table 1 into tables,
each with two variables. The sums of squares are computed the same as for a simple
randomized block test. It is to be noted that totals are used for the variable con
cerned. For this reason, the sums of squares obtained must be divided by the number
of basic plots included in the respective totals (to reduce the variables to a single
plot basis) in order for the common correction factor to apply.
The combined data for a comparison of varieties and stations are given in table 2.
TThe roman numeral refers to the line in Table 8.
198
Table 2. Total Yields grouped for Varieties and Stations
Barley
Variety
Total
U.Farm
1280.2
Station
Waseca Crookston Grand Sapid s Total
Manchuria
2365
260.8
Glabron
2+8.1+
290 . 2
Velvet
251 . h
2.81.6
Wis. #38
30U .
32^.9
Pea tl and
239.9
317.9
IkiJ.k
2.1.8.9
1772
218.1
211.2
2637
1089.1
152.2
868. h
107.5
323 . 3
ljk.6
925.7
137.2
977.3
187.1
1008.6
758.6
il603.3
These data are taken directly from the righthand column of table 1
of the sums of squares is carried out as follows:
The computation
Total
(X 2 V J  (Sx)2 = 1,129,020.73
176,586.^1
6 N 6
= 183,1701216  176,536^2lil = 11,583,6975
Varieties = S(x 2 v )  jSx) 2 =■ h, 261, 251.19  176,586*^2^1
2k N 2\v
= 177,5521329  176,586i+2'M = 9657088
Stations * 3(x 2 s)  (3x) 2 = 5, 577, 329*97  176,586^2'+!
30
30
= 185,910.9990  176.5861+2^1 = 9,52^.57^9.
Varieties x Stations = Total  (Varieties + Stations)
 11 ..583.6975  10,290,2337 = 1.293.1+153
The values for varieties, stations, and varieties x station total are included in
table 7, where the steps for computation are indicated. The interaction values are
included in table '8.
The sums of squares for the other factors are computed in a similar manner, the data,
being given in Tables 3> K, 5? ' :n 'l 6, with the results included in fable 'J.
In table 3 d^e given the data for comparisons of varieties and years, the yields at
the four stations being totaled.
Table 3.
Total yields grouped
r 'or varieties and years.
Zear
Variety
1932 1935
Total
Manchuria
3I46 . h
roc n
863.4
Glabron
3I45 . k
)lT7 Q
4 ! ( • ~>
8253
Velvet
335.5
5^0 . 2
925 . 7
Wis. #38
3 C )0,. 3
6175
9773
Peatland
kke.i
561 . 9
1008 = 6
Total
1883.8
2719.5
1+603,3
199
In table 4 are assembled the data for comparisons of blocks and stations, the block
totals for the two years of each station being added.
Total
Table 4.
Total yields
of
blocks and stations
Stations
Block
U. Farm
Waseca
Crookston
Grand Rapids
Total
I
II
III
3756
422.6
482.0
5379
494.4
443.1
385.2
368.3
3556
226.5
276.5
255.8
1525.2
1561.6
1516.5
1280.2
1475.4
1089.1
758.6
4603.3
In table 5 are the totals for comparison of blocks and years. This table is assem
bled from the totals at the bottom of table 1.
Table 5 Total yields of blocks and years
Block
I
II
III
Year
1932
1955
628.4
657.7
5977
896.8
903.9
918.8
Total
1525.?
1561.6
1516.5
Total 1885.8 2719.5
One other table is necessary, that of stations and years,
46033
Total
1280.2
Table 6
Total yields of
stations and years
Station
Year
U. Farm
Waseca
Crookston
Grand Rapids
Total
1952
1955
4697
310 . 5
608.4
867.0
457.O
632.1
348.7
409.9
I883.8
2719.5
1475.4
10 39.1
758.
4603 . 3
The calculation of the sums of squares for the complete analysis can be performed
vith the least difficulty and confusion when the steps are carried thru in a routine
manner. Many of the calculations are given in table 7. The remainder follow easily
and logically,
200
Table 'J. Calculation of sums of squares.
Total of Calc. No. of Single _ Sum of Key
Varlate Squares from Varia plots in (Sx) Squares to
table bles each tot. N table
squared squared 3
4 5 • 1*4 6 ' 57
S (x 2 ) 200,879.35 1 120 1 2OO.879.35OO 176.586.^1 2Jf,292.0259 ■ III
S (x) 5,577,329.97 2 k 30 185,910.9990 " . 9;324.57 i +9 II
S (x 2 y ) 10,9^,382.69 3 2 60 I82,li06.3782 " '• 5 : ,819. 95^1 HI
S (x 2 v ) ^, 261, 25I.I9 2 5 2fc 177,552.1329 " 965.7086 IV
s (x\) 7,064,601.85 4 3 ko 176,6150462 " 28.6221 I
s (x 2 sy ) 2,897.377.01 6 8 15 193,156.4673 " 16,572,0432
S (x 2 J 1,129,020.73 3 20 6 138,170.1216 " 11,533.6975
To
S (x 2 , rv ) 2,206,62761 3 10 12 183,885. 63^2 " 7299.2101
S (x 2 :, q ) 1,871,824.37 4 12 10 l87,lo2.i+370 " 10,596.0129
S (>: 2 bv ) 3,650,180.03 5 6 20 i82,509.00]5 " 5 ; 922. 5774
B(x% aJ ) 5^3,855.03 1 h0 3 197,951.6767 ,! 21,3652526
S(x 2 b3y ) 97^., 322. 9^ 1 24 5 194,864,5860 " 18,278.1619
A notation found to be very convenient in practice is to let S(x L  J be the sum of the
squares of the individual plots. The station totals are designated x g , the variety
totals x v , etc. The totals for varieties at the separate stations are designated x.< rp .
varieties in different years by x^r, etc. These are given in table 7. In column 1
of this table are the sums of the squares of the varlate s concerned and in column 2
is given the tabic from which they havj been computed. Thus, S(x^) is calculated Iron
the 4 station totals of table 3 S(x5) is calculated Prom the variety totals of
table 2. The value of S(x 2 s ) is the sum of the rjquarcs of the 20 yields .for each
variety at each station separately in table 2.
Column 3 of table '( gives the number of figures squared under column 1. Column ''
gives the number of single plots contained in each figure squared. Column 5 is
simply columns 1 divided by 4. This is necessary to reduce the sums of square s to a
single plot basis throughout. The key numbers refer to that sum of squares in the
complete analysis of variance given as table 8.
The sums of squares for total, blocks, stations,, years and varieties are transferred
directly from table 7 "to table 8. The interaction sums of squares can be obtained
from table 7 by subtraction.
The sum of squares for Interaction of varieties x stations will be found by subtrac
tion of the sum of squares for varieties and stations separately from the suma of
squares opposite S(x 2 o ) in table ?■ e.g.
201
11,583.6975 (19 D.F. )
 965.7088 ( k P.P. for varieties) IV
9,324. 5749 ( 5 DF. for stations) II
1,293.^138 (12 D.F. for varieties x stations) V
Since there were 20 figures used to obtain S(x^ ) there would be 19 degrees of free
dom. The interaction degrees of freedom are obtained by subtraction, e.g., 19 
(4 + 3) = 12. All otner first order interactions are obtained in the same manner.
The second oraer interaction of varieties x stations x years is obtained by subtrac
tion from the sum of squares opposite S(x2 va ) in table 7 the sums of squares for
varieties, stations and years separately and their first order interactions in all
possible combinations. Thus:
21,365.2526
965.7088
9,32^.57^9
5,819.95^1
1,293.^138
513.5^72
1,4275142
(39 D.F.)
( 4 D.F. for varieties) ,— IV
( 3 D.F. for stations) . II
( 1 D.F. for years) III
(12 D.F. for varieties x stations) V
( 4 D.F. for varieties x years) VI
( 3 D.F. for stations x years) VIII
2,020.5396 (12 D.F. for stations x years) VII
The sums of squares for the main effects and the first order interaction for the com
putation of the second order interaction are taken from table 8 opposite the appro
priate key number. The interaction of blocksx stations x years is obtained in a simi
lar manner.
The complete analysis is now carried out in table 8, the error sum of squares being
obtained as a remainder.
Table 8. Complete analysis of variance
Key
No,
Variation due to:
D.F,
Sums of
Squares
Mean
Square
F Valued
I
II
III
IV
Blocks
Stations
Years
Varieties
2
3
1
28.6221
9,324.57^9
5,319.9541
9657038
14.3110
3,103.1916
5,8199541
241 . 4272
162.85**
304 . 92**
.12.65**
V
VI
VII
viii
IX
X
. XI
XII
Error
Interaction of:
Varieties x stations 12
Varieties x years 4
Varieties x stations x years 12
Stations x years 3
Blocks x stations 6
. Blocks x years 2
Blocks x stations x years 6
(Blocks x varieties 8)
(Blocks x varieties x stations 24)
(Blocks x varieties x years 8) 64
(Blocks x varieties x 3ta.x yrs.24)
1,2934133
513.5472
2,020.5396
1,4275142
1,242.8159
74.0012
360.6795
1,221.5546
107.7345
128.3868
168. 3783
4758331
207.1360
37.0006
60.1132
19.0868
5.65**
6 . 73**
8.82**
24 . 95**
10 . 35**
1.94
3.15**
XIII
Total
119 24,292.9259
**Exceeds the 1 per cent point in Snedecor's table of "F".
^For comparison with error.
202
In practice, it is unnecessary to keyout the complete analysis as given in table 8.
The variation due to blocks., blocks X stations, blocks x years, and blocks x stations
x years should be grouped as one quantity, being designated as "Blocks within sta
tions and years" or "Blocks within tests" (for 16 D.F.). The reason for this is
readily apparent. The blocks are numbered I, II, and III arbitrarily. Block I at
University Farm has no relation to Block I at Wasoca or any other station. Thus, it
is an error to regard blocks as a factor that occurs at several definite levels (3
in this case). The correct procedure is therefore to compute the block sums of.
squares for each experiment and combine them to present in the final analysis. The
analysis of variance may then be presented as in table 9
Table 9 Analysis of Variance (in summary form)
Variation
Sums
■ Mean
due to :
I) F.
Squares
Square
FValu..>
Blocks within Tests
16
1,706.1187
106.6324
5 . trqttH
Stations
3
9,324.5749
3,108.1916
162 . 85**
Years
1
5,819.95^1
5,319.9541
30 4 . 92.**
Varieties
4
965.7088
241.4272
12. 6 5**
Interactions :
Varieties x stations
12
1 ; 293 4138
107 7845
5.65**
Varieties x years
k
515.5472
128.3868
6.75**
Varieties x stations x years
12
2,020.5396
168.3783
3, 82**
Stations x years
3
1,427.51^2
475,8381
24 . 93**
Error
64
1 . 221 . 5546
19.0868
Total
119
24,292.9259
The analysis may be summarized still further in case the investigator is not inter
ested in the variation due to stations, years, and stations x years. He may group
these factors into variation due to "tests within stations and years" or simply
"tests" (for 7 B.F.). The variety factor and its interactions would be given in de
tail because it has definite biological significance.
V . Sum s of Squar es in Sim ple vs . Complex Experiments
It will be useful at this stage to relate the complete analysis in Table i
simple randomized bloc]': tests computed for each of the 8 tests separately.
with the
The sums
of squares for total, blocks, varieti
.es, aid. error are
siven in table 10 for the 8
tests.
Table 10 . Sums f
squares calculated
from the tests 8'
iparately
Sum of squares
Sum of squares
Sum of squares Sum of squares
Test Year
for total
for blocks
for varieties for error
U. Farm  1932
867.2973
450 . 90 73
3 75. 61 06 41.5894
U. Farm  1935
1031.7133 '
251 • 6253
506 . 3600 273 . 7280
Waseca  1932
I907.I36O
471 . 3960
1196.3293 2394107
Waseca  1935
1203.5600
131.1480
993.8400 78.5720
Crooks ton 1932
487.8733
80 . 8333
.. 252.6266 154.4134
Crookston 1935
807.3360
49.2040
5953227 162.3093
G. Rapids 1932
905.5573
179.6353
536.1640 189.7080
G.' Rapids 1935
509.9093
92.1293
336.4560 31,3240
Total
7,720,8825
1,706.1185
. 4,793.2092 1,221,5548
■ 203
It is noted that the sums of squares for error for the 8 separate tests adds to
1,2215548. This agrees with the sum of squares for error of oable 8 (1,2215546)
the discrepancy "being due to dropping of decimals. There were 8 D.F. for error in
each separate test or 8 by 8 = 64 in all tests. These same 64 D.F. were used for
error in the complete analysis in table 8. The error used in table 8 is, therefore,
simply the average error of the separate tests.
The sums of squares added for blocks, in the 8 tests, gives 1,706.1185 (see table 10)
Addition of the sums of squares for blocks, blocks x stations, blocks x years, and
blocks x stations x years from table 8 gives a total of 1,706.1187, which agrees al
so. Further comparisons are given in table 11.
Table 11. Comparison of degrees of freedom and sums of squares of the 8 separate
tests with the complete analysis of table 8.
Variation
due to:
Calculated from 8
separate tests
Calculated from the complete analysis
D.F.
Sum of sq.
D.F.
Sura of Sq.
Key to table 8
Blocks
Varieties
Error
(8)(2)
(8)00
(8)(8)
= 16
= 32
= 64
1,706.1135
4,793.2092
1,221.5548
16
52
64
1,706.1187
4,795.2094
1 , 221 . 5546
I, IX, X, XI
IV, V, VI, VII
XII
Total
(8) (HO
=112
7,720.8825
112
7,720.8827
I,IV,V,VI,VII,IX,
X,X1,XII
From the above table the analogy between the separate analyses of variance for each
test and the complete analysis is clear. The 112 D.F. for total in table 11 is the
total sums of squares within test s. When the 7 D.F. between tests (i.e., stations =
5, years = 1 and stations x years = 5) are added, the full 119 D.F. is obtained. The
same is true for the sums of squares.
VI. Interpretation of the Data
The manner in which the data can be interpreted will now be illustrated. From table
8 it is 3een that the variance (mean square) due to varieties compared with variance
due to error exceeds the 1 per cent point. Therefore, some of the varietal differ
ences are significant. The mean yields for varieties, computed from Table 1, are
given in table 12.
20k
Table 12. Mean yields of 3 plots of each variety, average yields for both years and
average yields of varieties for all tests.
Year
1932
1933
Manchuria
26.9
31.9
Variety
Glabron
Velvet
Wis. #38
Unive r s ity Farm
~ 36.8 26.8
_ 46*0 37.0
38.0
63.3
Peatland
>1.9
Mean yield
39^
41.4
4 1.9
50.7
40.0
1932
Mean Yield
335
53.5
43. 5
Waseca
377
4875
37^
56 . 5
46.9
2D
.0
54.2
U7.H
L.'
53.
1932
1935
Mean Yield
35.0
4o.o
36
5
Croo ks ton
26.2 *
32.9
295
32.1
40. 6
36 IT
25,
35
359
,_52,0_
44.0
1932
1935
Mean Yi
22.1
cJ. O
25.4
Grand Rapid s
14.4
21. 4
179
JC. c.
26.0
?9.1
20.7
251
22.9
26. o
31.2
Mean for all
stations
~*>6.2
^
58.6
40 . 7
The varianc e due t o error was 1 90868 (table 8). The standard error of a single plot
would beVl9<0868 = 4.369 bu. Since 24 plots are involved in the variety averages
Qno
DU ,
for all stations, the standard error of the mean of 24 plots is 4.36 9 = 0.89
The standard difference between two such means would be 0. 892V 2 = 1.26 bu. With 64
D.F. for error one may accept twice the standard error of the difference as a level
for odds of approximately 10:1 against the chance occurrence of a difference of
(2)(1.26)  2.52 bu.
Since the mean yield of Peatland for all stations and years was 42.0 bu., any variety
that differs from it by more than 2.5 bu. may be judged as probably significantly
lower in yield, on the basis of these tests alone. On this basis Manchuria, Glabron
and Velvet are significantly lower in yield than Peatland.
The interaction of varieties x stations was also significant (table 8). A first order
interaction is essentially a difference between two differences. The mean yield of
Peatland at University Farm, for an average of both years, was 10. 7 bushels less than
the yield of Wisconsin #38 (50,7  40.0 = 10.7). The mean yield of Peatland at Grand
Rapids exceeded the yield of Wisconsin #38 by 3.3 bu. (31.2  22.9  8.3). The ques
tion then is whether these two differences art. significantly different. This differ
ence between two differences will be given by Wisconsin #38 minus Peatland at Univer
sity Farm less Wisconsin #38 minus Peatland at Grand Rapids, or (50, 7  40,0) 
(22 .9  31.2) = I9.O bu. The standard error of this "cross difference " will be
/ l9.086S x g"x 2 = 3.367.
V 6
MB
205
I t may als o be computed as follows: The standard error of the mean (o~f ) is equal to
/19.0868 = 1.784 bu. , since 6 plots are contained in each mean. The standard error
of the difference between two differences then is 1.784 V2'VF= 3567 bu. Twice this
is 7.13 bu. and any "cross difference" that exceeds this value is expected to occur
less than once in 20 trials by random sampling alone. The cross differences for
Peatland and Wisconsin #38 at University Farm and Grand Rapids greatly exceed 7*13
bu., being 190 bu. It is clear, therefore, that these two varieties responded in a
differential manner at University Farm and Grand Rapids as an average of 1932 and
1935 Other significant cross differences could be found in the same way.
Significant interactions of varieties x years could be determined by application of
the general procedure outlined above. Since only two years are involved these inter
actions of varieties x years can have very little practical significance.
While the second order interaction of varieties x stations x years was also, signifi
cant, it is of secondary interest. This significant second order interaction merely
means that certain differential responses of varieties x stations were not constant
in different years. To illustrate the types of comparisons which must be made to
show this, take th^ means of Glabron and Velvet at University Farm in 1932 and 1935
separately and the same yields at Grand Rapids. Then: f (56. 8  26.8)  (46.0  5"J0)]
£(l4.4  32.2)  (21.4  26.0) ] = 34.2 with an error of / J9OQ68 x 2 x 2 x 2 or
7.13 bu. V " 3
Since the difference of 34.2 exceeds (2)(7.13) = 14.26 bu. it is obviously significant.
For a complete understanding of a complex analysis, of which that given in table 8 is
an example, one further comparison can bu made. Suppose that V, S and Y are designat
ed to represent variance due to varieties, stations and years and V x S, V x. Y and
V x S x Y the interaction variances. Then one may determine whether the variance due
to:
V > V x S > V x S x Y
> Error
V > V x Y > V x 3 x Y
by means of the "F" test. When the variance due to varieties significantly exceeds
the interaction, varieties x stations, there is evidence that varietal performance
generally was consistent enough to demonstrate that some varieties were the bust in
all stations, as an average of the years in which tests were made. When the variety
variance significantly exceeds that of varieties x years, one may conclude that, as
an average of all stations, some varieties were consistently better in yield in the
different years.
Further, when the interaction of varieties x stations significantly exceeds varieties
x stations x years, it is plain that the differential response of the varieties at
the separate stations were sufficiently similar in the different years to warrant the
conclusion that these d.ifferential responses may be permanent features of these local
ities.
Unless the variance for varieties significantly exceeds that of varieties x stations
or varieties x years, no general recommendations can be made for the entire state or
for future years. To make such recommendations the stations (of which tests were
made) are considered as random samples of all places 1n the state and. the years in
which tests were conducted must be considered as a random sample of all future y^ars.
It is only when the number of stations and. years can be considered an adequate sample
of all possible places and years that worthwhile predictions can be made for all
places in the state and for future years. (Sec Summerby, 1937). •
206
VII. The Homogeneity Test
The question may be raised as to whether the data afforded by the several experiments
are sufficiently alike to assume that they may have resulted from a single' population.
In case this is true, the data from the experiments may he consolidated and. analyzed,
as one complex experiment. Homogeneity tests have been suggested by Sne decor (1937)
and by Stevens (1936).
The formula may be explained as follows:
Let n  the number of experiments.
n  1 = the number of degrees of freedom between experiments.
M = the number of degrees of freedom for error within an experiment.
e : the sum of squares for error in a single experiment.
v = the observed, variance
V  the theoretical variance
L = the Lexis ratio
The observed variance of the sums of squares due to error for all experiments is:
▼ = S(e  e) 2 = S(e 2 )  (Se) 2 /n
n  1 n  1
The theoretical variance, where the total., S(e), is assumed, to be the population of
error sums of squares, is as follows:
2M
5(e) rp or 2_ " Sje)'
M (nl_)j m _ n .
o
The Lexis ratio (L ) is the ratio of the observed to the theoretical standard error,
so that its square is L 2  v/V. When the ratio is greater than one. the series of
sums of squares due to error is called supernormal. When "L" is less than one,, it is
called subnormal .
A certain degree of supernormal ity or subnormal! ty can be attributed to cnance. The
limits for significance can be determined by the X, 2 test, viz.,
X 2 = (nl^L 2 or (nl) (v)
(vO
When X 2 corresponds to a probability of less than 0.05, the series is too supernormal
to admit that they resulted from, a single population. AX 2 that corresponds to a
probability greater than 0.°5 indicates that the series is too subnormal : f 'or consoli
dation of the data.
The homogeneity test can be applied to the data on' the barley yield trials as compiled.
in the separate tests in table 10 :
Test Year Gums squares due to error
U. Farm 1932 '+1.5894
U. Farm 1935 . 275.72a')
Waseca 1QJ2 239A107
Waseca 1935 78. 5720
Crooks ton 1932 . i5JJ.il 3**
Crookston 1935 162.8093
Grand Lap ids 1932 I89.708O
Grand Rapids 1935 • 81.32^0
n  8, (n1) = 7, M = 8. S(e) 1221. 55^8
nH
207
The sinus of squares for the eight "error sums of squares" is as follows:
S(e 2 ) = S [(41. 589M 2 + (2737280)2 + (81.3240) 2 ]
= 233,100.8233
(Se) 2 /n = l86,5?4.5773
S(e  s) 2 = S(e 2 )  (Se) 2 /n = 46,576.2460
v = S(e  5) = 46,576.2460 = 6,6537494
n1 7
V = 2M
S(e)"
nM
= 16
1221.5548
L (B)(8) .
2 = 16 (19.0868) 2
= (16) (364.3059) = 5828.8944
X 2 = (nl) f v) = (7) (6655.7494) = 7. 9906'
V 5828.8944
When theX2 table is entered for 7 degrees of freedom, P = 0.3335
Therefore, the data are sufficiently homogenous to permit the calculation of one
generalized standard error for all tests.
VIII. Transformation of Percentage Data
Some types of discrete data cannot "be comMned to provide a valid estimate of a
generalized standard error. This applies particularly to some forms of percentage
data wherein each variate represents a certain number of observations of a given type
or condition out of a total number of trials or cases (N). The variance of a single
variate of this type is pqN. It is clearly dependent upon p, the estimated ratio of
existence of the type or condition in question, as well as upon N. Bliss (1937),
Salmon (1938), Cochran (1938), Clark and Leonard (1939), and others have recognized
that each variate in discrete data of this kind does not have the same opportunity to
contribute equally to a general experimental error.
(a) The Angular Transformation
R. A. Fisher has supplied a mathematical transformation for such data which will
equalize the estimated variance of each variate so that it is functionally dependent
only on N, the total number of trials. In this transformation, each estimate of p is
replaced by sin 2 Q whence,
0= Sin "7p or 1/2 Cos" 1 (l2p)
This transformation must be applied to discrete data of this type so that the analy
sis of variance may be valid. However, it is of little practical importance when the
percentage values are between 30 and 70. Bliss (1937) has compiled a convenient
table for the transformation of percentage values to angles, the latter being measured
in degrees (See Table 5, appendix).
(b) Classification of Percentage Data
The type of discrete data, rather than its expression in percentages, determines
whether or not the transformation should be employed. The types of percentage data
are classified by Clark and Leonard (1939) as follows: (1) Continuous data from an
experimental study may be expressed as percentages when each variate is divided by an
arbitrary constant value, whereby each variate becomes a percentage of some standard
'<& ■ a
or average. Clearly such a procedure merely transforms the unit of measurement. Per
centages of this type should be treated statistically exactly as though the data were
in their raw form. For example., yield data might "be expressed in percentage of the
check instead of actual yield in pounds. (2) Continuous data are often expressed in
percentages to show concentrations. This type of percentage is very common. Some
examples are: seed purity given by weight of pure seed/ total weight of seed, leafi
ness given by leaf weight/ total plant weight, protein content given by weight of
protein/ total weight; sugar content given by weight of sugar /weight of root. etc.
Such concentrations should not, as a rule, be subjected to any transformation to
equalize the variance. (3) The third type of percentage is where the original data
are discrete, being based upon a determinate number of trials or cases (N). The
transformation, p = sin^fr should be applied to this type where it is desired to con
struct a generalized standard error. Illustrations of this type are as follows:
Germination percentages given by number of seeds germinated/ total seeds, disease
percentages given by number of plants diseased/ total plants, etc.
Referenc es
1. Bliss, C. I. The Analysis of Field Experimental Data Expressed as Percentages.
Plant Protection Bui. 12, U.S. 3. P. (Leningrad). 1937
2. Clark, Andrew, and Leonard, Warren H. The Analysis of Variance with Special
Reference to Lata Expressed as Percentages. Jour. Am. Soc. Agron., 31:5566.
1939^
3 Cochran, W. G. Some Difficulties in the Statistical Analysis of Replicated
Experiments. Emp. Jour. Exp. Agr. , 6:157175 1933.
4. Fisher, R. A. Design of Experiments. Oliver and Boyd. 2nd cEd. pp. 7576
1937
5> Goulden. C. H. Methods of Statistical Analysis. John Wiley, p. 139 1939
6. Immer, F. R. , Hayes, H. EC. and Powers, L. R. Statistical Determination of
Barley Varietal Adaptation. Jour. Am. Soc Agron., 26:403419, 1934.
7. Paterson, D. D. Statistical Technique in Agricultural Research. McGrawHill
pp. 5865, 6667, and 205208, 1.939
8. Salmon, S. C. Generalized Standard Errors for Evaluating Bunt Experiments for
Wheat, Jour. Am. Soc. Agron.. 30:647663. 1938.
9 Snedecor, G. W. Statistical Methods. Collegiate Press, Inc. pp. 196197. 1937
10. Stevens, W. L. Heterogeneity of a Set of Variances. Jour. Gen., 33:398399.
1936.
11. Summerby, R. The Use of the Analysis of Variance in Soil and Fertilizer Experi
ments with Particular Reference to Interactions. Scl. Agr., 17:302311. 1937.
Questi ons f or Dis cus sion
1. What are the advantages of a complex experiment over separate single tests?
2. As a matter of design, would it be necessary to have all varieties in all locations
in each year? Why?
3 Why does the total sum of squares for the simple tests fall short of the total sum
of squares for the complex experiment? What would make them check?
4. How would you interpret a significant interaction such as, for example, varieties
x stations?
5 Explain why a first order interaction is essentially a difference between two
differences.
6. What is a homogeneity test? Why should it be made?
7 Under what conditions should percentage data be transformed to degrees of an angle
to admit valid use of a pooled estimate of error?
209
Problems
1. The yields in "bushels per acre for five spring wheat varieties tested in 3 rando
mized "blocks for 3 years are given below. (Data from F. R. Immer).
Block Number
I II III
Tot.
Block Number
Block Number
I II III
Grand
Variety
I II III Tot.
Tot. Tot.
Thatcher
Ceres
Reward
Marquis
Hcpe
U. Farm  1951
17.0 20.0 19.7 56.7
16.1 18.9 20.5 55.5
21.1 25.1 21.8 66.0
15.4 20.9 18.4 54.7
20.3 21.0 14.2 55.5
U. Farm  1932
33.6 37.7 31.2 102.5
29.7 30.0 55.9 95.6
24.1 26.9 29.8 80.8
26.3 31.3 29.8 87.9
28.1 25.4 51.5 85.O
U. Farm
32.4 3^.3 373
20.2 27.5 25.9
29.2 27.8 30.2
12.8 12.3 14.8
21.7 24.5 23.4
 1955
104.0 2632
736 224.5
87.2 234.0
399 182.5
69.6 210.1
Total
89.9 1039
Waseca 
26.8 356
292 32.4
23.3 22.8
26.2 28.8
22.6 21.0
94.4 288.2
1931
26.4 86.8
26.0 87.6
18.5 64.6
25.5 78.3
24.2 67.8
141.8 151.8 153.2 451.8 116.5 126.41516 574.51114.3
Waseca  1932 Waseca  1935'
Thatcher
Ceres
Reward
Marquis
Hope
22.3 20.7 25.5 67.0
24.3 26.2 26.7 772
27.2 24.9 24.6 76.7
27.8 26.5 24.0 73.1
24.0 25.7 23.3 71.0
28.5 30.1 30.1
15.6 14.5 14.5
13.0 22.4 25.2
15.4 6.4 4.9
23.0 29.0 25.5
88.7 242.5
44.4 209.2
•65.6 206. 9
24.7 181.1
82.3 221.1
Total 128.1 138.6 113.4 585.I 126.1 121.8 122.1 570.0 105.5 102.4 998 5057 1060.8
Thatcher
Ceres
Reward
Marquis
Hope
590
54.6
52.5
51.4
Crookston  1951
30.4 lOf.O
376
37 4^
31.3
26.4
337 105.7
29.3 936
30.5 88.3
Crookston  1932
Crookston
25.1 15.2
51.1 19.5
23.1 22.8
20.1 19.2
20.8
20.9
19.8
15.5
591
71.5
657
54.8
27.0
15.4
16.8
5*
24.2 17.5
11.0 11". 5
16.4 14.6
5.9 8.4
1935
~£o\7 234.8
579 215.1
47.8 207.1
17.7 160.8
27.8 50.6 29.4 87.3 19.5 25.2 20.8 65.5 18.0 18.5 150 51.5 204.6
Total 165 5 1653 1538 482.4 116. 9 101. 9 973 316.6 32.6 75.8 67.0 225410224
Total 383.5 405.8 566.6 1155.7 584.3 5755 573.11153.4 502.4 502.6 29814 905. 4 51975
(a) Calculate the analysis of variance for the complete study.
(b) Test the significance of the different mean squares compared with the error
variance; using the F test.
(c) Compare Thatcher with Ceres, as an average of all tests, using the standard error
of the difference.
(d) What would be the standard error for testing the significance of the interaction
between Thatcher and Ceres in 1952 and 1935> as an average of all stations? Make
the proper test of significance.
(e) Do the same, as under (d), for comparing Thatcher and Ceres at University Farm
and Waseca, as an average of all years. Is this interaction signigicant?
(f) Calculate the sums of squares for blocks,, varieties, error and total for each of
the 9 separate tests and add the different components for all 9 tests. Compare
these sums of squares with appropriate combinations in the complete analysis of
variance table.
210
2. Test the data in problem 1 for homogeneity.
3 The relative infection in different varieties for 5 bunt collections were as fol>
lows (Data from Salmon . 1^38):
Variotv
Bunt Oro Rid it Alb it Turkey
Collection (l)^ (g) [l] _[2j_ £U (2) (l) (?)
(Pet.) (Pet.) (Pet.) (Pet.) (Pet.) (Pet.) (Pet.) (Pet.)
1 0.0 0.9 6.3 3.9 0.0 0.0 8.3 6.5
2. 2.5 3.6 8.7 2.2 92. U 90.5 89.O 8'4.3
3 1.5 0.0 6.0 0.7 93.7 90.1 33 6.0
k 1.5 6.3 k.l 3.1 1U.0 lj.5 81.7 87.2
5 0.6 1.7 3.9 3.6 U,2 32 7,5 2.U
(a) Transform those percentage data to degrees of an angle and compute the analysis
of variance for varieties, replicates, and collections.
(b) Compute the data without the transformation and compare the results with (a).
vThese numbers refer to replicates,
' CHAPTER XVIII
THE SPLIT PLOT EXPERIMENT ^ ".
I. Use of Split Plot Experiments
Sometimes it is an advantage to use relatively large plots for one series of treat
ments and subdivide these whole plots into a number of subplots to superimpose a
second series of treatments. This type of design, called the splitplot experiment,
was first proposed by Yates (1933.* 1935) It is particularly useful in spacing
tests with crop plants, some fertilizer trials, and in cultural studies. Le Clerg
(1937) used this type of experimental design to ascertain the effect of 5 fertilizer
mixtures (main treatments) on the seedling stand in plots sown with treated and un
treated seed (sub treatments) . Goulden (1939) gives a more complicated splitplot
design in which he studied the incidence of rootrot on wheat varieties, kinds of
dust for seed treatment, method of dust application, and efficacy of soil inoculation
with the root rot organism.
The split plot design provides a more critical comparison of the subplot treatments
than it does for the wholeplot units. This is due to the larger number of replica
tions of the small units which, in turn, provide a larger number of degrees of free
dom fo r error . Paterson (1939) advises that the less important treatment effects
be allocated to the whole plots and the more important treatment effects to the sub
plots in order to obtain the maximum precision where it is most desired.
The splitplot design leads to two or more errors. To simplify the computation, all
treatment values should be expressed in subplot units.
II. Data usca for Computation
Two designs are outlined below together with the method of computation. These data
are from a corn uniformity trial conducted at Waseca (Minnesota) in 1933 by C. \J .
Doxtator. They are for yield in pounds for the central two rows of fourrow plots
12 hills long. For purposes of calculation, it was supposed that these data were ob
tained from 10 hybrids which are designated 1,2,3 10. It was supposed further
that these varietal plots had been split into three parts to test the yield of those
crosses obtained from F^, F 2 , and F* generation seed. These are designated a,b, and
c, respectively. The yields in the tables that follow are in the same order as in
the field. The hypothetical hybrids and generations were superimposed on the data
by random arrangement , . '"
III. Sub treatments Randomized withi n Main Plot (Plan A)
The field arrangement of the plots is given below. The 10 hybrids are assumed to
have been planted in rows of 36 hills, using Fj seed for 12 hills, F 2 seed for 12
hills and F^ seed for the remaining 12 hills. The order of the hybrids in the field
is random and the three generations of seed for each hybrid are planted in a random
order within each long. row.
Ha
r brid Number
3
n
2
1 .
6
7
10
9
k
5
c
a'
a
c
b
c
b
c
b
c
a
b
c
b
a
b
c
a
a
b
b
c
b .
a
c
a
a
b
c
i'This chapter is a modification of one prepared by Dr. F. P. Immer, University of
Minnesota, for his Applied Statistics course.
211
212
The. yields of each plot are given "below in table 1. Data from two "blocks are used
to illustrate the calculations.
Table 1. Yield of corn per 12 hill plot and sums of yield o f 36 hill plots.
Block I
Hybrid Number
Total
3
8
2
1
6 7
10
9
>±
_JL.
a
48
c
i+6
a
1+6
a
1+2
c b
43 )+7
c
1+8
b
46
c
1+6
h
1+9
c
a
b
c
b
a
b
c
a
a
1+6
^5
1+1+
1+6
45
49
45
48
43
49
b
b
c
b
a
a
a
b
c
^3
42
1+2
44
1+1+
47
45
47
47
48
T. 137
133
132
132
132
11+3
138
141
141
146 1373
b:
.ock II
Hybri
.d Number
Total
4
3
9
.?
1
7'
2
'
10
c
a
a
b
b
c
c
a
b
c
1+6
45
46
1+5
*3
48
44
44
hi
1+3
a
b
c
a
c
b
c
a
b
1+8
1+1+
U6
1+5
50
48
46
48
^3
b
c
b
c
a
a
b
b
c
a
1+2
1+2
It 4
^3
1+4
48
47
46
44
42
T.136 13.1
136
133
137
147
136
139
,28 1362
( a ) Calculation of Sums of S qiiar es
The analysis of variance is given in table 2
Table 2„ Analysis of Variance
Variation due to:
Blocks
Hybrids
Error (a)
D.F.
1
9
9
oums of Squares
2.8166
77.6833
8I.OI67
Mean Square
2.8166
8.63I5
9.0019
Plots of hybrids
:i9)
I6I.5166
7.2533
88.7667
40.6667
298.I833
Generations
Hybrids x generations
Error (b)
Total
2
13
20
59
3.6167
4.9315
2.0333
The total sum of squares is calculated from the squares of the 60 individual plot
yields as S( x 2 )  (Sx) 2 /N which, numerically is 125,151.0000  124,852.81.67 =
298.1833 (59 D.F.).
The sum of squares for blocks is I375 2 + I562 2  124, 8o2.8lo7 = 2 t 8l66 (1 D.F.)
30
213
Sum of squares for total plots of hybrids is calculated from the marginal totals for
hybrids in the above table. Thus,
157 2 + 135 2 j 128 2 12i<,852.8l67 = 161.5166 (19 D.F.)
3
To obtain the sums of squares for hybrids, generations and. the interaction between
them it is necessary to set up another table with the yields of the two replicates
of each treatment combined.
Generation:
Hybrid Number
Sum
1
2
3
if
5
6
7
8
9
10
a
86
9±
93
96
9^
92
97
89
93
87
921
b
87
91
87
89
9^
92
98
88
90
88
90^
c
96
86
88
92
91
87
95
92
9k
91
912
Sum 269 271 268 277 279 271 290 269 277 266 2757
Sum of squares for hybrids will be: .
269 2 ± 271 2 + 266 2 12it,852.8l67 = 77.6833 (9 D.F.)
6
Sum of squares for generations is obtained from
92 1 2 + 9042 + 9122 _ I2i+, 852. 8167 = 7.2333 (2 D.F.)
20
The sum of squares of the 30 yields in the above table will be equal to 86 ^ 9^ *
 — 91 2  12i+,852.8l67 or 173.6833 (29 D.F.) 2 '
Sum of squares for interaction of hybrids % generations will be:
173.6833 (29 D.F.)
7.2333 ( 2 D.F.) for generations
776333 ( 9. D.F.) for hybrids
88.7667 (18 D.F.) = sum of squares for interaction
(b) Errors to Test Significance
These sums of squares are brought together in table 2. The sum of squares
for error (a) are obtained by subtracting the sums of squares for blocks (1 D.F.) and
hybrids (9 D.F.) from "plots of hybrids" (19 D.F..). The sum of squares for error
(b) is obtained by subtracting "plots of hybrids", generations and hybrids x genera
tions from the total.
Error. (a) is an ordinary randomized block error and may be used to test the
significance of differences between hybrids .
Error (b) is obtained from the sum of the interactions between generations and
blocks within hybrids. Thus, a table could be arranged for the data from hybrid No.
3 (see table 1) as follows:
Block I
Block II
Generation
a
b
c
lf8
k 1 ?
^3
kk
k6
k2
21k
The interaction of blocks x generations, for 2 D.F., could "be used as error for this
simple comparison. However, a table similar to the above could be set up for each
hybrid. There would be, then, 10 x 2 = 20 degrees of freedom for error. This is
what is used as error (b) . The mean square for error (b) is, then, the average
error of blocks x generat ions wibhin hybrids . It will be the legitimate error to
use for comparing differences between generations and testing the interaction of
hybrids x generations. In practice this sum of squares is obtained by subtraction.
IV . Sub treatments in Str dps Across Blocks ( Plan B )
The same yield figures are used in this plan as in Plan A. The location of the
hybrids is also the same. Instead of randomizing the generations within the 1 plots
for each hybrid as in Plan A, the generations are now considered planted in long
strips crosswise of the entire block. However, randomization of the generations in
the different blocks is used. The field plan is given below.
Table 3.
Yields
of corn
plots
and the
field
arrangeme
nt of
these
plots.
Block I
Hybrid
Numbt
iV
Total
3
B
2
1
_ _£. ....
7
10
Q
k
c.
a
1*8
a
1*6
a
1*6
a
1+2
a
h5
a
kl
a
kQ
a
k6
a
J+6
a
i* 9
I+61
c
1*6
c
1*5
c
kk
c
1*6
c
1*5
c
kg
1 s
c
1*5
kQ
c
1*3
c
1*9
h63
b
43
b
1*2
b
1+2
b
kk
b
kk
b
^7
b
b
kl
b
b
1+8
kk')
Tot.137 133 132 132 132 lk$ 138 l'+l li+l ii*6
Block II
Hybrid Number
Total
k
3
9
5
1 7 2 8 6
10
b b b b b b b b b b
1*6 1*5 1*6 45 1*3 1*8 li1* 1*1+ 1+7 k'5 1*51
c ccccc ccc c
1*8 44 46 1*5 50' 51 1*8 46 1*8 1+3 469
a a a a • a a a a a a
1+2 k2 kk 1+5 1*4 1*8 1*7 1+6 1*1* 1*2 1*1*2
T0U.36 131 136 I33 137 l)+7 139 136 139 128 1362
The some plots are used here as in table 1. The hybrids occur ;ln the same order as
in the previous table, the only difference being the arrangement of the generations.
In table 3 the generations occur in strips crosswise of the blocks.
215
Table 4. Analysis
of variance from the data
of
table
3
Key number for
D ,F . and sums of
squares
Variation, due
• to
D.F.
Sum of
Square s
Mean Square
1
2
3 = 412 .
E locks
Hybrids
Error (a)
1
9
9
2.8166
776833
81.OI67
2.8166
8.6315
9.0019
= 715
Plots of hybrids
Blocks
Generat ions
Error (b)
19
161.5166
1
2.8166
2.8166
2
2
35.4333
16.2334
17.7167
8.1167
5
54.4833
19
1+
161.5166
51.6667
18
18
61.5667
23.1+333
3.4204
1.3019
8=5+6
10 = 1198J*
Plots of "generations"
Plots of hybrids
Deviation of genera )
tion plots from blocks)
Hybrids x generations
Error (c)
11
Total
59
298.1835
The total sum of squares (298.1835)* sum of squares for blocks, hybrids, error (a)
and total plots of crosses will be the same as under Plan A. The position of the
"generation" plots has "been changed, however, and the other sums of squares must be
recalculated.
As far as 'the test of the three generations a, b and c is concerned there are but .
six plots as given in the marginal total of table 3. The sum of squares for those
six "plots of generations" will be
l+6l 2 4I.65 2 ± 1+1+ 9 2 + 451 2 ± 469 2 ± 442 2  12l+ , 852 . 8167 = 54.4833(5 D.F.)
10
To obtain the sum of squares for generations and for interaction, a table is made up
by combining the two yields of each treatment .
Genera
tion
IT
Hybrid Number
5
~8~
"b
c
86
87
96
93
86
92
90
88
90
8b
93
96
92
93
9l+
Tot.
269
271
263 277
070
79
271
290
269 277
10 Tot.
87
95
92
90
90
903
91
95
86
93
83
900
93
100
91
91+
88
934
266 2737
The sum of squares for generations will be
354333 (2 D.F.)
9002
9542 . 12^,852.8167 
20
The yield figures in the above table are squared, i.e., So 2 + 93^ + 88 . The
sum is divided by 2, the correction factor then being subtracted. This gives
174.6833 as the sum of squares for bhese 29 degrees of freedom. The sum of squares
for the interaction, hybrids x generations will he: I7I+.6833  77.6833  35.4333 =
61.5667 (18 D.F.).
216
In table k it is noted that the comparison of hybrids is the same as under Plan A.
Error (b) will he obtained by the subtraction of blocks and generations from the
"plots of generations." It is seen from table 3 that the analysis of variance to
test the significance of generations involves only 6 large plots. The total yields
of these could be set down from the marginal totals of table 3 as:
Generations
a
b c
Total
Block I k6l
Block II hh2
M+ 9 1+65
k'jl 469
1375
1362
Total 903 900 93^ 2737
An analysis of variance of this 2by3 table would give the second section in the
complete anal.ysis of table h. Error (b) is appropriate for the comparison of dif
ferences between generations.
Error (c) is obtained by subtraction from the total the items listed in table k
opposite error (c). Error (c) is the second order interaction of blocks x hybrids
x generations., the degrees of freedom being 1 by 9 by 2 = 18. It was obtained in
table k by subtraction but has the above meaning. Error (c) is appropriate for
testing the significance of the interaction of hybrids x generations.
Since these were uniformity trial data no attempt will be made to determine signifi
cance of the different mean squares. In a practical experiment these tests are
carried out in the ordinary way, the appropriate errors given in the tables being
used .
Yates (1933) has discussed the above two designs rather fully.
Y. Comparison of Two Designs
Suppose the 10 hybrids and 3 generations of the seed of each (F]_ ., Fg and F*) had
been considered as simply 30 treatments and completely randomized within the blocks
without reference to split plot arrangements. The analysis of the data would have
taken the form:
Variation due to: D.F.
Blocks 1
Hybrids 9
Generations 2
Hybrids x generations 18
Error 29
Total 59
The degrees of freedom for error given above (29) is equal to the sum of the degrees
of freedom for errors (a) and (b) under Plan A and the sum of degrees of freedom for
errors (a), (b) and (c) under Plan B.
Plan 3 is the same as Plan A insofar as precision of tests of the hybrids is con
cerned. It differs from Plan A in that precision for the comparison of generations
is sacrificed in order to obtain greater precision for the interaction.
The design of an experiment will depend entirely on what element of the treatments
the highest degree of precision is desired, When the primary emphasis is to be
217
placed on the Interactions, at the expense of higher errors for the main 'effects,
Plan B Is to be preferred. When the main effects are of major interest either the
complete randomized block or Plan A are to be preferred.
In practice the relative differences in magnitude of the different errors under
Plans A and B will depend on the dimensions of the blocks . In this case the blocks
•were 1+0 rows wide, or lAO feet, and the 36 hill rows of hybrids were 126 feet; long.
Consequently the "generation" plots tended to be closer together than the most dis
tant hybrids, in the same block.
Plan A is particularly applicable to studies of space relationship between plants in
relation to yield. In a recent study of the effect of spacing on yield of soybeans,
conducted^by the Division of Agronomy and Plant Genetics, U. of Minnesota, Plan A
was found 00 be admirably suited to the test. The soybeans were planted in k row
plots, the rows being 16, 20, 2k, 28, 32, and ho inches apart. Then, the soybeans
were planted at k different rates within each spacing, being l/2, 1, 2, and 3 inches
apart within rows . The only oa.sy way to lay out such a test was to plant the plots
of different width rows crosswise of the regular 132foot series. The k different
rates of seeding were then randomized within these long rows, the ultimate plots,
being 33 feet long.
Plan A could be laid out as follows also, using the same notation as employed in
table 1.
Hybrid H umber
3
•
8 :
2
: etc .
a
: c
: b :
c : a : b :
b
: a :
c
: etc.
Here the hybrids are planted in groups of 3 plots with the three generations in a
random arrangement within each hybrid plot but they occur side by side instead of
end to end. By this plan it is obvious that the comparisons between generations
(a, b, and c) will have a lower error than comparisons between hybrids (1,2,3 ).
The data from this arrangement would be analyzed exactly in the same manner as given
under Plan A.
VI. RandomizedBlock vs. SplitPlot Experiments
The relative efficiency of randomizedblock and splitplot experiments was studied
on uniformity trial data with sugar beets by Le Clerg (1937) both in the field and
in the greenhouse. He compared the magnitude of the variance of the subplots with
in main plots in the splitplot design with the variance of subplots within blocks
in the randomized block arrangement. The variance for subplots within main plots
in the split plot design was markedly less than that for subplots within blocks in
the randomized arrangement. The splitplot design was 71 per cent more efficient
in one set of uniformity data and 53 per cent more efficient In another. For com
parisons of main plots within blocks there was a decrease in efficiency by the use
of the splitplot arrangement. Similar results were obtained for greenhouse trials,
altho less marked.
References
1. Goulden, C. H. Methods of Statistical Analysis. John Wiley, pp. 151159 1939
2. LeClerg, E. L. Relative Efficiency of RandomizedBlock and SplitPlot Designs
of Experiments Concerned with Daraningoff Data for Sugar Beets'. Phytopath.,
27:9^2  9I+5. 1937.
218
!
J. Paterson, D» D. Statistical Technique in Agricultural Research. McGrawHill.
pp. 209214. 1939.
4. Yates, Fi The Principles of Orthogonality and Confounding in Replicated Expert
merits. Jour. Agr; Sci., 23:1081^5, 1933.
3. ComplexExperiments. Jour. Roy. Stat. Sec. Suppl., 2:l8l223. 1933.
Questions for jD is cuss ion
1. What is a split plot design? Where used to advantage? List at least 3 situa
tions .
2. Explain the differences in field layouts that lead to two and three errors.
3. Under what conditions would you use Plan "A"? Plan "B"?
4. Compare the relative efficiency of splitplot and randomizedblock designs
super imposed on uniformity trial data.
li'roplems
The following data are from a randomized "block experiment with "split plots'' designed
to test the differences in yield of soybeans planted at different spacings between
and within rows. Four row plots were used, one row being harvested for hay and one
for seed.
(A) Yield of soybeans in "bushels per acre
Block Width Block
No. of rows 1/2" 1" 2" 3" Total Total
I 16" 25.1 21.3 22.3 22.1 Q0.8
20" 21.8 22.7 22.2 22.8 89.3
24" 21.9 21.8 21.2 20.6 8.3.5
28" 21.2 20. 4 20. k 17.9 79.9
32" 20.7 20.0' 16.3 20.0 79.0
ko^ 19.3 18.3 i?V3 16.3 JIJS ^v^o
II 16" 25.2 I9.9 22.1 22.7 89.9
20" 21.9 21.3 22.1 22.9 88.2
24" 19.7 I9.8 20.1 19.8 79.4
28" 20.8 21.2 18.8 20.6 81.1+
32" 18.3 20.7 17,5 16. 4 73.1
4o" 18.3 18.2 1 9.8 15 .j2 72.4 484.4
III 16" 15.7 21.6 22.9 " 20.3 80.5
20" 22.0 20.4 22. 4 20.7 835
24" 23.5 20.7 20.7 20'. 5 87.4
28" 21.5 19.9 20.3 20.9 82.8
32" 22.0 ' 19.3 13.1 17.8 7 7 2
4o^ 20_ ! 3 16.4 17 ^3 13.3 729 486.3
IV 16" ' 23.8 29.0 12.3 23.5 88.6
20" ■ 27.0 21.2 20.5 20.7 89.4
24" 23.3 20.0 22.5 19.8 33.6
28" 22.3 21.3 22,7 13.9 83.6
32" 23.9 18,4 20.7 18.7 31.7
4o» 19.9 17.8 16.9 18.3 73  1 04,0
Total 322.6 491.8 479.8 476.8 1971.0 1971.0
219
(B) Yield of dry hay In tons per acre
Block
Width
Block
No.
of rows
1/2"
1"
2"
3"
Total
Total
I
16"
2,91
2.59
2.41
2.7^
10.65
20"
2.96
2.35
2.31
2.10
9.72
24"
2.3^
2.30
2.21
2.23
9.08
28"
2.59
2.47
2.16
2.10
9.32
32"
2.21
2.12
2.05
1.90
8.28
40"
2.24
1.90
1.82
1.79
7^75
54.80
II
16"
2.85
2.42
2.45
2.31
10.03
20"
2.42
2.48
2.31
2.27
9.48
24"
2.40
2.19
2.29
2.08
8.96
28"
2.48
2.22
2.30
2.08
908
32"
2.32
2.10
2.23
2.06
8.71
40"
2.34
2.07
1.76
1.78
7.95
54.21
III
16"
2.81
2.61
2.65
2.25
10.32
20"
2.66
2.52
2.78
2.52
10.48
24"
2.57
2.41
2.28
2.15
939
28"
2.03
2.22
2.39
2.01
8.65
32"
2.68
2.21
1.97
1.96
8.82
40"
2.13
2.09
1.84
1.96
8.02
5568
IV
16"
2.83
3.10
2.12
2.38
10.43
20"
3.27
2.71
2.33
2.42
10.73
24"
2.71
2.31
2.22
1.97
9.21
28"
2.52
2.53
2.24
2.09
9.38
32"
2.37
2.29
2.20
1.85
8.71
4o"
2.10
2.13
1.92
2.06
8.21
56.67
Total 60.74 56.34 53.24 51.04 221.36 221.36
The actual field arrangement of plots in this experiment, in block number III
was as follows: The plot arrangement in the other "blocks was randomized in a similar
manner.
Width of rows
 16"
32"
28"
40"
24"
20" 3" 1" 1/2" 2"
1. Analyze the data on yields of soybeans in bu. per acre.
(a) Calculate the complete analysis of variance. Test the significance of the
different mean squares, compared with the appropriate error variances, by
means of the F test.
(b) Determine the significance of the difference between 20" and 32" rows by means
of the standard error.
Spacing
wi
thin
rows:
1/2"
1"
3"
2"
1"
3"
2"
1/2"
y
1/2"
a.
2"
2"
1"
1/2"
3"
1"
3"
2"
1/2"
220
(c) Determine the significance of the difference "between 1/2" and. 2" spacings by
means of the standard error.
2. Analyze the data on yields of soybeans for tons of dry hay per acre in a similar
manner.
3 Key out the degrees of freedom for a split plot experiment (two errors) for 3
spacings, 4 blocks, and 3 widths of rows.
4. A splitplot experiment was designed to determine the effect of seed treatment on
the stand and ultimate yield of dryland corn planted at 3 different dates., a,, b,
c. Each plot consisted of 2 subplots., the seed being treated (l) with an organic
mercury compound in one half,, and untreated (U) in the other half. There were 3
me/in plots in each block. All treatments were randomized. The field design of
Block I was as follows:
!
T ' U
t
T
T ' U
t
U ' T
i
b
c
a
i
The yield data for the 6 blocks of the experiment were as fallows in bushels per acre:
Date
Seed
Bl
ock
Planted
Treatmi
snt
1
2
•2
h
3
6
Total
a
U
2.3
4.6
3.4
2.3
58
33
24 . 2
T
2.3
4.7
4.2
3.6
5.0
4.6
24.9
b
U
h.3
M
33
6.1
4.5
4.0
26 . 5
T
5.1
6.1
31
4.3
5.3
5.9
29.8
c
U
2.7
1.4
2.3
38
2.9
59
17.0
T
2 .
.1.3
1.3
fc7
3.4
15
15.2
(a) Calculate the complete analysis of variance.
(b) Determine the significance between treated and untreated seed, and also between
planting dates.
CHAPTER Xn
CONFOUNDING IN FACTORIAL EXPERIMENTS
I. Factorial Experiments
The randomizedblock and Latinsquare designs are widely used in field experiments,
"both "being very efficient for simple studies. However, there are situations in ex
perimentation where a large number of varieties or treatments are to be compared at
two or more levels. The factorial experiment is useful in such situations. Suppose
that three fertilizers, Nitrogen (N), Phosphorus (P), and Potash ,(K) are to he test
ed at two or more levels. The classical method of approach would he to vary the two
levels for each element only one at a time, i.e., the investigator would set up
separate experiments to test each element alone at its respective level. The single
factor could then he studied under controlled conditions at each of the two levels.
To test these factors simultaneously in the same experiment, would permit one to
study the effects of different amounts of one fertilizer on the others in all combi
nations. Thus, a wider "base of inductive reasoning is provided. The experimental
argument is also strengthened by the larger total number of plots in the test. (See
Fisher, 1955).
Goulden (1937) describes a factorial experiment as one made to study simultaneously
various treatment factors. Thus, an experiment designed to study at the same time
rate and depth of seeding of a cereal crop would be a factorial experiment in which
two factors, rate and depth, are represented at two or more levels. The study of
interactions is an important consideration in such an experiment. The introduction
of factors is limited by space and cost of experimentation.
Suppose a fertilizer test is to be conducted \#ith II, P, and K at two different rates
each. The rates can be designated by subscripts so as to give the eight possible
treatment variants as follows: •*
NqPoKo, NiPqKo, NoPiKo, NqPoKi, N1P3.K0, NiPqKi, NqPiKi, and NiP]Ki
The degrees of freedom, i.e., the number of comparisons free to vary, may be keyed
out as follows:
Variation due to
Degrees freedom
Bemarks
Nitrogen (N)
Phosphorus (P)
Potassium (K)
N x P 2
N x K
P x K
N X P x K
1 )
1 )
. 1 )
1 )
1 )
1 )
Li
Main effects
First order interactions
Second order interactions
Total
■'•Note: The subscripts and 1 represent the two fertilizer levels.
The symbol (x) denotes interaction and not a variable as heretofore.
221
222
II . Data for Computation of Factorial Exper i men t
The computations for the Analysis of Variance for such a factorial experiment will
he illustrated with uniformity trial data. Four complete replications will he used.
The uniformity data on crested wheatgrass were furnished by Dr. B. M. Weihing. The
plots are combined as 8low plots, 16 feet long, with rows 6 inches apart. Thus,
each plot is k hy 1.6 feet in size. The yields are given in grams of airdry field
cured hay. The uniformity trial data follow.
Tahle 1. Uniformity Data for Crested Wheatgrass
Blocks
Plot No. I II III IV
(gnu) (gm.) (gnu) (gnu)
1 5135 3175 ^05 3750
2 I4725 3980 1+575 3920
3 1+600 W+20 3910 1+175
•>+ 1+955 I+580 ^065 3280
5 3210 3970 3510 . 3190
6 3670 1+255 ] +305 3573
7 3735 3665 3993 3530
8 3965 1+315 I+030 2900
III. Computation as Sim ple Randomised Block Experiment
The eight treatments will first he superimposed on the crested wheatgrass yield data
for a randomized hlock test. 
Tahle 2. Yields of Crested Vlieat gra ss in Eandoialzod Blocks
Tre,
atmont
Eeplication
N
F
K
HP W£
PK
NPK
Totals
I
II
III
IV
3210
'3970
1+305
3530
U955
3175
1+1+05
^175
5135
1+1+20
1+573
3920
1+600
3980
3910
3280
5963 3733
i+255 3665
1+030 3510
2900 3575
3670
^515
3995
3190
1+725
1+580
I4O65
3750
3I+OI+5
32560
32795
28320
Totals
15015
16710
18050
15770
15150 1.1+535
15170
17120
I27320
The sums of squares for "blocks, treatments, total, and error are computed in the
ordinary manner. The Analysis of Variance can hu summarized as follows:
Variation
D.F.
Sum
Squares
Mean
Square
"]?"
Value
due to
Oh served
5 Pet . Point
Blocks 3 2,303,556 767,832 3.91 307
Treatments 7 2,628,237 375,1+62 1.91 2.57
Error 21 k, 125,107 12^^'ik ,
Total 31 ~ '9,056", 900 "'
The hlock effect removed is just enough to he statistically significant. Treatment
effects are within 'the limits of error since the data are from, a uniformity trial.
■Htfote: The same randomization for treatments is used here as in the confounded experi
ment to he mentioned later.
223
The crested wheat yield data will now he considered from the standpoint of confound
ing. This process is expected to accomplish several things: (1) A greater amount
of the variability due to soil heterogeneity should he removed because more and
smaller blocks will be used; (2) A chance to examine the second order interaction,
N x P x K, will be forfeited; and (3) The reduction of experimental error in this
manner should sharpen all treatment and interaction comparisons.
IV. Confounding in a 2 by 2 by 2 Experiment 1
A few terms must first be made clear before the analyses are made.
(a) Explanation of Terms
Every effort is made to maintain orthogonality in an experiment. Yates
(1933) defines orthogonality as follows: "Orthogonality is that property of the de
sign which ensures that the different classes of effects shall be capable of direct
and separate estimation without any entanglements." Thus, orthogonality is ensured
in a randomized block experiment by the very nature of the design, i.e., each block
contains the same kind and number of treatments. Non orthogonality is introduced
when some of the plots in one or more of the blocks are lost . Special methods may
be required to separate treatment and block effects.
Nonorthogonality is sometimes deliberately introduced in factorial experiments that
involve a fairly large number of combinations . This process is called confounding.
The purpose is to increase the accuracy of tie more important comparisons at the
expense of the comparisons of lesser importance.
(b) Confounding the Second order Inte raction
The second order interaction (N x P x K) in this experiment may be considered
the least important. Certainly, it would be difficult to interpret in terms of fer
tilizer practice, even though it were significant. Suppose it is desired to con
found the one degree of freedom for this interaction with blocks. To accomplish
this, it is necessary to determine the distribution of treatments in the blocks in a
manner so as to confound this one treatment and, at the same time, leave the others
intact .
Algebraically, the treatment effects can be represented as follows:
Nitrogen (N) =
(*1
" N ) (Px
+ P )(K 1 + K )
Phosphorus (P) =
<*1
*oN»i
+ N )(% + K )
Potassium (K) =
(Ki
 K o)(%
+ N )(P 1 +P )
N x P
(»1
 * )(P1
 P )(K X +K >
N x K
(>1
 N ) (%
 K )(P 1 +P )
P x K
(Pi
?o)( K l
 K Q )(N 1 + N Q )
N x P x K
(Hi
 HjfcPi
 Pj(Ki  O
The last expression can be expanded as follows:
N x P x K = (N X  N Q )(P 1  PjjXKx  K ) =
(N X P K t I^A ♦ *£&  %?!%)  (XfJb  NiP^ + N^Kj. + N^K^ =
(N + P + K + KPK)  (0 + UP + PK + NK)
Then, the blocks could be divided as follows so as to confound the second order in
teraction with block effect:
N P K NPK NP PK NK
Sub block A Sub block B
^For more eomplicated designs see Yates (1933), Fisher (1935) and Goulden (1937).
22*+
The contracts "between the two sub blocks of each replicate will "be contrasts of the
second order interaction (N]i?iK]_ and N P K ) . This interaction will have "been con
founded with "blocks.
The sum of squares for the second order interaction will be given by: (See Goulden,
1957).
1/2 k ; (N + P + K + NPK)  (0 + NP + PK + NK)]
where k  number of plots represented in each total. The above sum of squares will
contain not only the second order interaction effect but also the block effect.
In this ca.se, blocks of four plots each 'have been used for error control instead of
blocks of eight (as would be the case in a simple randomized block experiment); and
only the second order interaction has been lost. The keyout for four complete
replications will be as follows:
Variatio n d ue t o Dec r ees o f free dom
Blocks 7
N 1
P 1
K 1
IxP 1
N x K 1
P X K 1
. error
Total
1 u
The treatments will be randomized in each subblock. The field arrangement and plot
yields follow:
Table 3. Field Plan with Plot Locations and Yields
S ub b lock A Sub block B
Replication Treatment Yield Treatment Yield
1
N Pl K
NlPlKi
NpPoKl
NiP K
Total
NlP K
N P Kl
NoPlKo
5155
J +725 .,
1+600
191+15
5175
3980
kkZO
i458o
NqPqKo
NoPlKi
KlPcICl
N1P1K0 ■
5210
3670
5735
396^
II
Total
N0P0K0
WlPlKo
NiPqK]
NoPlKi
]Jo30
3970
1+253
3665
^315
III
Total
Nl?oKo
N ?iK
NoPoKl
N1P1K1
16155
41+05
1,573
5910
M065
Totai
NiPoECi
KqPoKo .
NcPlKl
NlPlK
16265
5510
H305
:>9?o
1+030
IT
Total
UlPlKi
NqPxKo
NlP K
NpPoKi
16955
5750
3920
^175
3280
Total
N ?lKi
WlPpKl
NcPbKo
N1P1K0
15840
3190
3575
3530
2900
Total
15123
Total
13195
The yield data are summarize for main effects in Table h as follows
225
Table 4. Total Yields for Four Replications per Treatment
Ko
Kl
Sum
Ko
Kl
Sum
No Po
?1
15,015
18,050
15,770
15,170
50,785
55,220
Hi *o
16,710
15,150
1^,555
17,120
31,21+5
52,270
Total
55,065
50,9^0
64,005
Total
51,860
51,655
65,515
The yields for the various interactions are totalled below in Table 5:
Table 5. Total Yields for Interactions
Comparison
(a) N and K
( p o +
Pl)No
Nl
Ko
Totals
Kl
Sum
55,g65
51,860
50,940
31,655
64,005
65,515
Total
64,925
62,595
127,520
(N +
N l> P o
*1
Ko
Kl
Sum
(b) P and K
51,725
55,200
50,505
52,290
62,050
65,490
Total
64,925
62,595
127,520
(*o +
N l
p o
p l
Sum
(c) N and P
50,785
51,245
55,220
52,270
64,005
63,515
Total
62,050
65,490
127,520
The sums of squares for the experiment are given in table 6. The sum of squares for
"blocks, N, P, K and total can be entered from table 6. The sum of squares for N x P
is obtained hy the subtraction of 7505 + 374,112 (N + P) from 445,745 which is S(xp).
The result would be 62,128. The* sums of squares for N x K and for P x K are obtained
in a similar manner.
Table 6. Calculation of Sum of Squares
Total
Divide
Corrected
Symbol
Tahle
Sum Squares
by
(Sx) 2 /N
Sum Squares
D.F.
S(x2)
5
517,224,100
1
508,167,200
9,056,900
51
S(x2)
5
515,954,112
4
503,167,200
5,786,912
7
S0§)
4
8,130,795,250
l6
508,167,200
7,505
1
S(x 2 )
4
8,156,661,000
16
508,167,200
374,112.
' 1
S(x)
4 •
8,133,589,650
16
508,167,200
169,653 ■
1
S(x 2 )
NP
5
4,068,887,550
8
508,167,200
.443,743
1
S K 2 J
HE
s
4,067,676,450
8
503,167,200
292,356
1
S(x 2 )
PK
5
4,069,752,750
8
508,167,200
55.1,895
1
The analysis of variance, together with the obtained and theoretical "F" values are
presented in Table 7.
226
Table 7 Analysis of Variance
Variation
D.F.
Sum Mean
Squares Square
"F" Value
due to
Obtained 5 pet.
point
Blocks 7 5,786,912 826,702 5.87 2.66
N 1 7;503 7,503 18.76 2}+3.91
P 1 37^,112 37^,112 2.66 KM
K 1 169,653 169,653 1.21 KM
NxP 1 62,128 62,128 2.27 2V3.9I
NxK 1 115,200 115,200 1.22 2U3.9I
PxK 1 8,128 8,123 17.32 2U3.9I
Error 13 2,553,2 64 L'iQ,757
Total 31 9,056,900"
It is noted that the mean square for error has "been decreased materially in the con
founded experiment as compared, to that in the simple randomized "block experiment . In
the former, the mean square is 1^0,737 while in the latter it is 196, K^K . It is also
to he noted that more of the variability due to soil heterogeneity has been removed
from the experimental error and drawn off in block effect which now appears as highly
significant.
The real value of confounding as a means to bring out more closely significant treat
ment effects and interactions is not evidenced in this illustration because uniform
ity data have been employed. The confounding design is purely artificial.
V. Partial Confounding i n a 2 by 2 by 2 Experiment
The above procedure resulted in the complete sacrifice of the second order inter
action, but it may be argued that the experimenter has taken too much for granted.
He may overcome this difficulty by partial confounding, i.e., confounding different
interactions in different replications. Goulden (1937) states that the results are
used from the blocks in which the particular effects are not confounded in order to
recover a portion of the information desired. The fertilizer test used as an example
can be partially confounded and at the same time recover a portion of the information
on all the comparisons. Four replications will be required for this purpose. In
each replication, one degree of freedom can be confounded with blocks for one of the
interactions. There are four interactions, viz., N x P, N x K, P x K, and N x ? x K.
The algebraic relations stated previously can be used to determine the treatments to
place in each subblock to gain the desired effect.
Bub Blocks
Interaction Algebraic Relationship A
NxP = (N]_  N )(P 1  P )(K 1  K ) = (N+P+MK+PE)  (0*HP+K+HPK)
N x K = {Nil ~ N oH K l * ^( p l + ? o) c (N+K+KP+PE)  ( 0+P+HK+HEK)
p x K = (Pi'  P )(Ki  *c)(Nl + N ) = (P4K+IIP+ME)  (0+N+PE+NPE)
N x P x K = ( Nl _ N C )(P 1 . p^(K }  K ) = (N+P+K+NPK)  (O+NP+PK+NK)
The treatments within each subblock will be randomized. Table 8 gives the field
design together with the plot yields in grams for the fertilizer trial superimposed
on crested wheatgrass uniformity trial data.
227
Table 8. Field Arrangement and Yields In Partially Confounded Experiment
Sub
block A
Sub
block B
Replication
Treatment
Yield (gm. )
Treatment
Yield (gm.)
I
P
5135
3210
(N x P confounded)
PK
i+725
K
3670
NK
46oo
NP
3785
•
H
Total
^955
19415
NPK
Total
3965
14630
II
N
3175
NK
3970
(NPK confounded)
P
3980
NP
4255
K
1+420
3665
NPK
U580
PK
4315
Total
16155
Total
16205
III
(P x K confounded)
NP
44o?
NPK
3510
P
4575
N
4305
NK
3910
PK
3995
K
1+065
4030
Total
16955
Total
15840
IV
(N x K confounded)
N
3750
P
3190
PK
3920
NPK
3575
NP
4175
NK
3530
K
3230
2900
Total
15125
Grand Total =
Total
127,520
13195
The treatment totals required for the computation of the sums of squares are arranged
in Table 8 for the totals of the four blocks, and for the omission of each replica
tion.
Table 9« Treatment Totals Required for Calculation of Sums of Squares
Treat
All
Minus
Minus
Minus
Minus
ment
Replications
Replication
I Replication II
Replication III
Replication IV
. I3805
10595
10140
9775
10905
IT
 16185
11230
13010
11880
12435
P
16880
11745
12900
12305 .
13690
K
15435
II765
11015
11370
12155
NP
16620
12835
12365
12215
12445
NK
16010
11410
12040
12100
12480
PK
16955
12230
12640
12960
13035
NPK
15630
II665
11050
12120
12055
(1)
(2)
(3)
(*0
(5)
(6)
The sums of squares can be computed as follows for the treatment effects (for 1 d.f .)
N = 1/2 k [(N + NP + NK + NPK)  (0 + P + K + PK)] 2
P = 1/2 k [(P 4. NP + PK + NPK)  (0 + N + K + NK)J 2
K = 1/2 k E(K + NK + PK + NPK)  (0 + N + P + NP)] 2
N x P = 1/2 k [(N + P + NIC + PK) (0 + NP + K + NPK)] 2
228
N x K = 1/2 k [(N + K + KP + PK)  (0 + P i M + NPK)] 2
P x K = l/2 k [(P + K + MP + NK)  (0 + N + PK *■ NPK)] 2
H x P x K = l/2 k [(N + P + K + NPK)  (0 + NP + PK + NIC)] 2
For example, the interaction N x ? ia calculated from the replications in which it is
not confounded, i.e., from Column 3, Table 9 Note that k = 12.
N x P = l/24 [(11230 + 117^5 + 11410 + 12230)  (10595 + II763 + 12835 + 11665)] 2
= l/24 [ 1+6615  46860] 2
= l/2'+ [245] 2 = 60025/24 = 2501.04
Similarly,
N x P x K = l/24 [(13010 + 12Q00 4 11015 +■ 11030)  (101.40.* 12365 4 12040 *• 12640)] 2
N x P x K = l/24 [(47973  47185)1 2 = 1/24 [790] 2
= 624,100/24 = 26,004
The main effects are calculated from all the replications; i.e., k  16. The calcu
lation for N is as follows:
N = 1/32 [(16135 + 16620 +■ 16010 + I563O)  (13803 4 16880 + 15435 + I6953)'] 2
= 1/32 ['64445_ 63075] 2  i/32 [1370] 2
= l,37b,900/32 = 58,653
The total sura of squares is calculated from all plot yields in all replications of
the experiment, i.e., 32 plots. The block sum of squares is computed from the 8
block totals. The ordinary method of computation is used.
The analysis of variance can be set up as follows:
Table 10. Complete Analysis for Partially Confounded 2x2x2 Experiment
Variation
Sura
Mean
due to
D.F.
Squares
Square
Blocks
N
P
K
7
1
1
1
5,
786,912
58,653
675,703
9,112
326,702
53,653
675,703
9, 112
N x P
N x K
P x K
N x P x K
1
1 .
1
1
30,817
65,626
26,004
2,301
36,8.17
65, 626
26'. oo4
Error
17
2,
393,572
l4o,9l6
Total
31
9^
056,900
Obtained '
Value
5 Pet. Point
5
•Of
2
.40
k
.79
15
.46
3
•33
2
.15
5
.42
2.70
243.91
4.43
243.91
243.01
243.91
243.91
243.91
In this experiment, information is obtained on the main effects and on all interact
tj.ons, including the second order interaction. However, there is a loss of onefourth
the information on each of the interactions, due to the fact that the replication in
which an interaction was confounded was omitted in the calculation of its sum of
squares. The error is of approximately the same magnitude as that for the experiment
in which N x P x K was completely confounded.
229
References
1. Fisher, R. A. Design of Experiments, pp. 96I37. 1935.
2. Goulden, C. H. Methods of Statistical Analysis. Burgess Publ, Co., pp. 107120.
1937.
3. Wiebe, G. A. Variation and Correlation in Grain Yield Among 1500 Wheat Nursery
Plots. Jour. Agr. Res., 50:331357. (Source of Data). 1935.
k. Yates, F. The Principles of Orthogonality and Confounding in Replicated Experi
ments. Jour. Agr. Sci., 23:1081^5. 1933.
5. Yates, F. Complex Experiments. Suppl. Jour. Roy Stat. Soc, 2:1812^7. 1935.
Questions for Discussion
1. What is a factorial experiment? Give an example.
2. Under what conditions may a factorial experiment he used?
3. What is meant "by the term "orthogonality"?
Give an example of an orthogonal experiment.
k. Explain the use of the term "confounding". What is done in confounding? Why?
5. Suppose a secondorder interaction, N x P x K is to be confounded. How can this
he done "by design?
6. What is partial confounding? How does it differ from confounding?
Problems
Some uniformity data presented by Wiebe (1935) on wheat yields in grams per row are
presented below as they occurred in the field:
Plot
Blocks
No.
I
II
III
IV
1
670
690
785
6U5
2
685
790
770
665
3
660
825
960
750
h
705
805
860
635
5
610
720
705
615
6
6^0
735
805
665
7
690
855
905
700
8
715
765
9^5
820
1. Calculate these data as a randomized block experiment using the 8 fertilizer treat
ments given in the text example.
2. Design an experiment so as to confound the second order interaction, N x P x K.
Carry through the complete analysis. Compare the results with those obtained in
problem 1.
3. Design an experiment to superimpose on these data so as to partially confound the
second order interaction (N x P x K) . Carry through the complete analysis. Com
pare the results with those obtained in problems 1 and 2.
CHAPTER XX
SYMMETRICAL INCOMPLETE BLOCK EXPERIMENTS
I. Incomplete Block Teste
It has "been shown (Chapter 19) that greater accuracy is obtained in factorial ex
periments when certain degrees of freedom for the higherorder interactions are con
founded with "blocks, especially when the number of combinations is large. In varie
ty trials it is sometimes desirable to test a large number of varieties in a single
experiment. To compare them in an ordinary randomized block test leads to less
accuracy due to the large size of the blocks. Methods have been developed by Yates
(1936, 1937) to overcome this difficulty. The procedure is analogous to confounding
in factorial experiments in that the replications are divided up into smaller blocks
which are used as error control units. These small blocks contain only part of the
total number of varieties, hence the name "incomplete blocks".
Incomplete block experiments have been shown to give increased efficiency by Yates
(1936}, and Goulden (1937). Weiss and Cox (1939) found the lattice square arrange
ment to result in a gain of l';0 per cent on extremely heterogenous soil, but a loss
of precision of 3I>5 per cent on a very uniform soil.
One type of incomplete block experiment will be illustrated, i.e., the symmetrical
incomplete block where all possible groups of sets are used. The computation pro
cedure will follow closely that described by Weiss and Cox (1939) • For other types
of incomplete blocks, Goulden (1937? 1939) should be consulted. These include the
two dimensional quasi factorial with two groups of sets, and the three dimensional
quasi factorial with three groups of sets. An excellent discussion of the lattice
square design (quasiLatin squares) is given by Weiss and Cox (1939) who applied it
soybean variety test.
The computations will be illustrated with some uniformity trial data obtained from
Dr. R. M. Weihing on forage yields of crested wheat grass expressed in kilograms.
The plots consist of 3^ows, 15 feet long, the individual rows being 6 inches apart.
I I . Design of Symmetrical Incomplet e Block T ests
In order to determine the details of an acceptable design with regard to the number
of varieties and blocks to use, it is necessary to satisfy the condition that each
variety occur with every other variety in the same number of blocks. Suppose that
m varieties are replicated n times over a portion of the available blocks each of
which is to contain n' plots. For example, suppose that one considers the n plots
in which one certain variety occurs. The total number of plots contained in these
n blocks is obviously (n)(n'), of which n corresponds to the one variety under con
sideration. Therefore, there are (n)(n'n) = (n'l)(ri) plots available for the
other m  1 varieties in those blocks. To meet the above condition, these (n'l)(n)
plots must be distributed equally among the m1 varieties that remain. For this
reason, (n'l)(n) must either equal m1 or be a multiple of ja1 . Thus, it becomes
apparent that m1 = (n'l)(n) is a number chat must be; factorable, preferable into
two numbers of nearly equal size. This can be effected in two different ways.
(1) First, one may use m = k : , where k is an integer, from which m1 = k2l = (kl)
(k+1). From this, it would appear that the choice of design could be either kl =
n'  1 (i.e., n' = k whence n will be k + 1), or k + I * n'  1 (i.e., n' « k + 2)
in which case n will be kl. However, when in' is equal to the total number of
blocks, one must have ran = m'n' . Thus, it is clear that mn must be divisible by n 1 .
The first choice gives mn to be k^(k + 1) in which the divisibility is assured with
230
231
m' = k(k + 1) . The second choice gives ran a k£{k>lj, which generally would not he
n 1 k + 2
an integer. Thus, only the first choice is acceptable.
(2) Second, one may choose m = k 2  k + 1, from which to  1 = Is^k = k(kl). From
this relationship, it appears that one has a choice of design by the use of either
k  1 = n"  1 (i.e., n' = k) from which n will also be k, or k a n»l (i.e., n' =
k + 1) in which case n will be k  1. In the analysis of this situation, mn =
(l^k+Dk . The divisibility is assured with the result that m' = k^k+l. For the
k
second choice, mn = k2k+l , a value that is not generally divisible. Thus only the
n» k + 1
first choice is acceptable.
Therefore, it is obvious that designs of this nature can be constructed for m = k 2
where m varieties = 9>l6,25,36,49,64, etc. The k^k+1 type can be designed for
values of m = 7, 13, 21, 31> 43, 57 j 73 > etc. The structure of the arrangements is
rather fully discussed by Yates (1936), Fisher and Yates (I938), and by Goulden
(1937, 1939).
The first type, 1 = k^ s n'2, will be used to illustrate the process for a complete
ly orthogonal i zed 5 by 5 square. This will give a series of symmetrical incomplete
block arrangements.
1111
(1)
2222
(2)
3333
(3)
IjiiU^
w
5555
(5)
2345
(6)
3^51
(7)
4512
(8)
5123
(9)
1234
(10)
3521+
(11)
*U35
(12)
3241
(13)
1552
(14)
2413
(15)
42<>3
(16)
531^
(17)
1425
(18)
2531
(19)
3142
(20)
5432
(21)
15^3
(22)
215^
(23)
3215
(24)
4321
(25)
The explanation of the arrangement is taken directly from Weiss and Cox (1939) The
numbers in parentheses designate the varieties which are to be compared in the ex
periment. "These variety numbers are arranged in 6 orthogonal groups as follows:
Group
I
Group
II
Group
III
( rows )
( Columns \
(first Number)
1 2
3 k
5
1
6
11
16
21
1
10
14
18
22
6 7
8 9
10
2
7
12
17
22
2
6
15
19
23
11 12
13 14
15
3
8
13
18
23
3
7
11
20
24
16 17
18 19
20
4
9
14
19
24
4
8
12
16
25
21 22
23 24
25
5
10
15
20
_25
J_.
. 9
15
17
21
Group
IV
Group
V
Group
VI
(
second number)
12 20 23
1
(third number)
8 15 17 24
(fourth number)
1 9
1
7
13
19
25
2 10
13 16
24
2
9
11
18
25
2
8
14
20
21
3 6
14 17
25
3
10
12
19
21
3
9
15
16
22
4 7
15 18
21
4
6
13
20
22
4
10
11
17
23
5 8
11 19
22
5
7
14
16
23
5
6
12
18
24
In group I the variety numbers are copied from the rows of the square, each row of
the group specifying a block in the field. In like manner, the variety numbers in
the blocks of group II are taken from the columns of the square. In group III the
varieties in a block are specified by the numbers written first in the cells of the
square. Thus, the varieties in the first block are those corresponding to number
1 wherever it occurs first in tho cell; as examples, variety 1 is from row 1 column
1 of the completely orthogonalized square, variety 10 from row 2 column 5^ variety
232
lh from row 3 column h, .etc. "For group IV, the second numbers in the cells of the
square ere used to pick out the varieties . Thus, for the third "block the number 3
is located in row 1 column 3 (variety y) , in row 2 column 1 (variety 6), etc.
"This set of six orthogonal groups constitutes a balanced incomplete block arrange
ment: in the 30 blocks of 5 plots, each of the 25 varieties occurs 6 times, once
and once only with every other variety. The combination solution in the unreduced
form would require a prohibitive number of blocks^".
The field arrangement for this typo of symmetrical incomplete block design will be
illustrated with the crested wheatgraas data. There are 25 varieties arranged in
6 replicates with 5 varieties in each block. The 5 varieties ere randomized within
each block. The block and replicate arrangement in the field may be as follows cV
I
II
III
IT
V
VI
5b
6b
13b
20b
2Vb
27b
1ft
7b
lib
I8b
25b
29b
lb
10b
12b
1 .
lob
21b
26b
2b
8b
lift
•
19b
22b
23b
Jb
9"b
15b
17b
23b
30b
III. Statistical Analysis of Incomplete Block Data
The symbols used in the discussion follow:
m = number of varieties (25)
n' = number of plots per block (5)
n = number of replicates of each variety (6)
ia' = number of blocks (30)
N = inn = m ; n' = total number of plots (150)
X = n(n'  1) = number of times any 2 varieties occur together in a
m1 block (1)
E = 1l/n' = Efficiency Factor of Design, (5)
1T/m ' (6)'
Sx = Sum of all N experimental values ( 217. 79)
S 'x = Svsa. of n experimental values for any one variety.
V
S 'x = Sum of k experimental values for any one block.
B_
s^ = Error variance of a single experimental value,
^"b =_Q__= C
m n' 25 5
251 = 53, 130 >
5: 201
.vThe blocks (5b,Vb, etc.) were arranged consecutively for the analysis of the data
used in this problem, but they should be randomized (at least within replicates) in
an actual field experiment. The Roman numberals refer to replicates.
233
(a) Computation of Block Totals
The yield data for the incomplete "block experiment may he assembled. as shown
in Table 1 for the computation of the block totals. The numbers in parentheses
refer to "varieties". The forage yields of crested wheatgrass are expressed as
kilograms per plot .
Table 1. Plot Yields of the Symmetrical Incomplete Blocks Assembled for 25 Crested
Wheatgrass "Varieties" in 6 Replicates.
Beplicate
Set or.
Block
Plots in Block
Block
Totals
II
III
IV
VI
o
7
8
9
10
n
12
13
15
16
17
18
19
20
21
22
23
2k
25
26
27
28
29
30
(5
(10
(11
(20
(21
(1
(12
(23
(19
(15
(18
(15
(20
ik
(13
(l
(21*
(3
(*
(22
(15
(9
(19
(22
(16
(1
(2
(22
(10
(6
1.25
1.38
1.1*2
1.20
1.1+8
1.86
1.61*
1.81*
1.33
1.50
1.00
1.62
1.60
1.30
1.56
1.1*8
1.1*8
1.1*8
1.1*0
1.50
1.22
1.62
1.32
1.12
1.08
(2
(7
(1*
(18
(25
I.27 (16
1.92 (17
1.61 (3
l.Ol* (9
1.31* (25
(1
(2
(7
(12
(9
(20
(16
•(6
'(7
(5
(3
(2
(12
(20
(7
(25
(8
(9
(17
(18
1.52
1.1*8
1.1*0
1.32
1.1**
1.5 1 *
I.36
1.16
1.1*2
1.38
1.9 1 *
1.61*
1.72
1.5^
1.18
1.30
1.1*8
1.1*1*
1.31
1.1*1*
1.72
1.1*8
1.1*0
I.38
1.35
1.22
1.1*3
0.93
1.50
1.00
(3
13
17
23
21
(2
13
11*
10
11*
23
11
16
21
23
13
17
21
(8
17
11
21
(h
(5
(7
(20
(3
(11
(5
1.30 (3
1.1*1 (9
1.35 (15
1.59 (16
1.16 (22
1.66
1.32
1.1*7
1.16 (21*
1.02 (5
(6
(22
(8
1.81*
1.66
1.20
1.16
1.21*
1.21*
1.66
I.60
1.1*5
1.31*
(22
(19
(3
(8
(5
(12
(10
(11*
(15
(11
1.68. (2l*
1.29(25
1.26 (3
1.1*2 (13
1.61* (23
1.70
1.50
1.1*6
1.13
(13
(21
(15
(k
1 .08  ( 12
) 1.83
(1)
) 1.52
(6)
) 1.32
(12)
) 1.21
(19)
) 1 5k
(21*)
) 1.81
(11)
) 1.38
(7)
) 1.21*
(18)
) 0.92
w
) 1.22
(20)
) 1.96
(10)
) 1.78
(6)
) 1.29
(21*)
) '1.1*8
(25)
) "1.31
(17)
) 1.61*
(9)
) '1.62
(2)
) '1.56
(25)
) 1.51
(18)
) 1.58
(19)
) '1.9^
(1)
) '1.36
(18)
) 1.1*1*
(10)
) 1.7^
(6)
) l.oo
(i*0
)' 1.56
(19)
) 1.1*1*
(11*)
) 1.36
(16)
) 1.19
(23)
) 1.32
(21+)
1.61*
1.5k
1.1*6
7.25
1.19
6.68
1.22
6.51*
I.67
6.99
I.96
8.21*
1.65
7.63
1.32
6.32
0.99
5.55
1.72
6.68
1.92
9.52
I.83
8.55
I.29
6.89
I.5I*
7.05
1.53
6.80
1.62
•6.80
I.83
■8.21
1.81*
S.oi*
1.72
7.29
1.25'
7.17
1.81*
8.66
I.67
7.28
1.86
7.1*1*
1.55
7.^9
I.32
7.^9
1.60
7.30
1.56
7.55
1.1*1
6.1*8
1 . 10
6.09
1.31
5.79
Grand Total
217.79
(h) Computation of Variety Mea ns
In symmetrical incomplete block designs, a preliminary step is required to
obtain the sum of squares for varieties. Due to the fact that variety differences
are partially confounded with block effects, it is necessary to compute each variety
sum by a formula that involves both the yields of the plots planted to the variety
and the yields of the blocks in which the variety occurs.
23h
The first step is to accumulate the variety sums which are recorded in
table 2, column 2. The yields for each variety are collected from table 1. For
example, the total yield of variety 1 is:
S'  1.61+ + 1.27 + l9 ! + + 1.00 + 1,8k + 1.22 = 8.91
V
For each variety total there is also a sum of "block total (S'S'x) which is recorded
V B
in table 2. Since variety 1 appears in "blocks 1, 6, 11, 16, 21, and 25, S'S'x =
7.ih + 8.2I+ i 9.52 + 6.80'+ 8.66 + 7. 30 » 1+8,06 V b
Table 2. Computation of Variety Means for the Crested Wheatgrass Experiment with 2\
"varieties" in 6 Replications.
'arie"cy
Block tots .
Replicate
Variety for each
n'Sx 
Q
Variety
II III
IV
V
VI Totals S'x
V
V
tq t v
k> Q JL
25
Means
V B
Yields in Kg.
S'x
S 'S 'x
V B
Q
d
Sx/N
* d
1
1.61+
1.27
1.9!+
1.00
1.61+
1 i"» fj
8.91
1,3.06
3.51
0.1I+
1.31
2
1.52'
i.32
1.61+
1.83
1.1+8
1.62
9 .1+1
1+6.76
+0,29
+0 . 01
1.1+6
3
1.83
1.16
I.29
1.60
1.1+1+
1.1+6
O". (O
•'+3.21
+0 . 69
40 . 03
1.1+8
1+
1.30
0.99
1.33
1.30
1.1+2
1,19
7.53
1+0.99
3.34
0 . 13
1.32
5
1.25'
1.22
1.35
1.1+1+
1.61+
1.08
7.98
1+1.1+7
1.57
0.06
1.39
6
1.1+6
1.31
I.G3
1.1+1+
1.55
1,08
9.17
1+5,36
40,1+9
+0 . 02
1.1+7
7
1 .US
I.65
I.27
1.31
1.35
1.70
8.76
^3.83
0.05
. 00
1.1+5
8
1.1+1
1.24
1.1+6
l.3k
1.72
I.H3
P. ,<o
O . DC
1+I+.50
1.1+0
0 . 06
1.39
9
1.52
1.1+2
1.18
1.62
1.1:8
0.93
0.13
1+0 . Ik
40.61
40.02
1.1+7
10'
1.33
1.02
1.92
1.62
1,86
1.12
O. no
1+5.19
0.59
0 . 02
1.43
11'
L.l+2
1.96
1.20
1.38
1,29
1 i A
... j. 1^..
3.63
^2.3?
+0.80
+0 . 03
l.i+3
12
1.19
1.92
I.5I+
1 .61+
1.U0
1.32
9.01
kl.39
*3S6
+0 . 15
1.60
13
1.35
1.1+7
1.50
1.66
1.71+
1,56
9.28
4 3.^.0
+3 . 10
+0.12
1.57
ll+
1.1+0
1.16
1.81+
1.56
1.32
1.56
8.81+
1+1+.81
0.61
0.02
1.1+3
15
1.32
1.3^
1.61+
1.51
1.1+8
1.36
8.65
kk.3k
.1.09
0.01+
1.1+1
16
i .21
1.5k
1.16
1.1+8'
1.50
i.i+i
8.30
1+1+.01
2.91
0.10
1.35
17
1.59
1.36
1.33
1.60
1.68
1,50
9.36
U3.76
■i2 . 54
+0.10
1.55
1
1.32
1.32
1.86
1.72
1.67
1.00
8.89
1+3.24
1.21
+0 . 05
1.30
19 '
1.22
1.01+
1.78
1.25
1.1+8
1.60
8.37
1+2 . 53
0.63
0.03
1.1+2
20
1.20
1.72
1.81+
I.30
1 2.0
1.56
8.91+
1+1,95
+2.75
+0.11
1 . 56
21
1.1+8
1.66
1.21+
1A5
1.26
1.1+1+
8.33
I+1+.31
1.66
0.07
I.38
22
1.5^
1.38
1.96
1.36
1.1+0
1,32
9.16
1+5.23
+0 . 52
+0 . 02
I.V7
23
1.16
1.63
1.66
1 .21+
1.68
1,10
■8.47
1+2. 7I+
0.39
0.02
i.1+3
2k
1.67
0.92
1.29
1.62
I.9I+
1.31
8.75
1+2.07
. +1.68
+0 . 07
1.52
25 '
1.11+
1.3S
1.5k
1.81+
I.36
] 00
8.1+3
1+3.31+
0.9^
0 . 01+
1.1+1
Totals
)*
217.79^1088,95
0.00
0.00
\!/The sum of the S'x column (217 .79) is equal to Sx, while the sum of the S'S'x column
V V B
is equal to n'Sx. Therefore, the computations car. be verified: (5) (217. 79) =
1088.95.
235
For the computation of Q, the "block sums are subtracted from 5 times the variety
totals, i.e., Q = n'S'x  S'S'x
V V B
For example, for variety 1,
Q = 5(3.91)  bQ.o6 = 5.51
The Q value is then divided "by the number of varieties in the test (25) to give the
values for d in table 2. Thus, d is the deviation of a variety mean from the mean
yield of all the varieties in the experiment.
The best estimate of the variety means is Sx/N + d.
As an illustration, the mean of variety 1 is,
Sx/N + d = 217.79/150 + (0.1*0 = 1.31
In the variety means, consideration has been given to the effect of partial confound
ing of variety differences with block effects. They are the best estimates of ■ the
yield performance.
(c) Derivation of Sums of Squares
The sums of squares may now be computed. The correction factor is the square
of the total divided by the number of plots, viz.,
(Sx) 2 = (217. 79) 2 = klM2.kQkl = 316.22
N 150 150
The total sum of squares is obtained in the usual manner, i.e., by the addi
tion of the squares of each individual plot yield with the correction factor sub
tracted:
(1.25)5+ (1.52)2+ +(1.31)2316.22 = 8.23 
The sums of squares between means of blocks is obtained by the addition of
the squares of the block totals, these being divided by the number of plots which
make up each block total. The correction term is subtracted from this value.
(7.5*0 2 + (7.25) 2 z (579) 2
5
316 .22
fc.17
The sum of squares between means of varieties is obtained from each Q value
squared, added, and divided by N:
(3.51) 2 + (0.29) 2 + +(0.9*Q 2 = 0.56
150
The analysis of variance is presented in table 3.
Table 3« Analysis of Variance of Symmetrical Incomplete Block Design
Source of
Variation
D.F.,
Sum of
Squares
Mean
Square
F Value
Blocks
Varieties
Error
Total
29
2k
96
U.17
0.56
350
8.23
..1U58
0.0233
.0.0365
3. 9'+**
1.57
2%
The standard error of the plot yields is
s = /O.O365 = 0.19 kilograms
The standard error of the difference "between two of the corrected means will "be
2sf
n
n' + 1
n'
= /( 2) (O.O565) 6. = 0,12
IV. Efficiency Factor
The symmetrical incomplete "block design is less efficient than the complete ran
domized "block arrangement for equal numbers. of replications when the soil is homo
genous. This is because there has been no reduction in error variance duo to the re
duction of block size. The efficiency of the incomplete "block design as compared to
randomized complete blocks is expressed by the fraction, 1  1 /n ' , when the rcplica
1 ' 1/m"
tier, numbers in each arrangement are equal. In soils that are heterogenous the re
duction in block size usually more than compensates for the loss of information due
to the arrangement. Goulden (1937) concluded that an increase of precision of 20
to 50 per cent was obtained over the complete randomized block arrangement.
In addition to the doubtful value of the symmetrical incomplete block design on very
uniform soils, Weiss arid Cox (1939) advise that the design not be employed to com
pare varieties which have an extremely large range in yields, However, poor varie
ties are usually eliminated in preliminary trials. The symmetrical incomplete block
arrangement would provide a means to accurately determine relatively small differ
ences between select varieties.
Gculden (1939) gives a list of the n 1 and n values for different numbers of varieties
for which symmetrical incomplete blocks may be used:
Wo.
Varieties
13
16
21
25
31
h9
57
6k
73
No. Plots in
one block (n* )
h
k
5
5
6
7
8
3
9
No. Replications
for Each Variety (n)
6
6
8
8
9
9
References
1. Fisher, R, A. The Design of Experiments. Oliver and Boyd, 2nd Ed. pp. 100171.
1937 •
2. Fisher, R. A., and Yates, F. Statistical Tables for Biological, Medical, and
Agricultural Research. Oliver and Boyd. 1933.
3. Goulden, C. H, Efficiency in Field Trials of PseudoFactorial and Incomplete
Randomized Block Method. Can, Jour, Res. C, 13:2312^1. 1937.
^1 Modern Methods for Testing a Large Number of Varieties. Can.
Dept, Agr. Tech. Bui. 9. 1937 .
237
5. Goulden, C. H. Methods of Statistical Analysis. John Wiley, pp. 172202. 1939.
6. Weiss, M. G., and Cox, G. M. . Balanced Incomplete Block and Lattice Square De
signs for Testing Yield Differences among Large Numbers of Soybean Varieties,
la. Agr. Exp. Sta. Res. Bui. 257. 1939.
7. Yates, F. Complex Experiments. Suppl. Jour. Roy. Stat. Soc, 2:181247. 1935.
8. Incomplete Randomized Blocks. Ann. Eugenics, 7:121140. 1936.
9« A Further Note on the Arrangement of Variety Trials: QuasiLatin
Squares. Ann. Eugenics, 7:319332. 1937.
10. The Design and Analysis of Factorial Experiments. Imp. Bur. Soil Sci.
Tech. Comm. 33. 1937. '
Questions for Discussion r
1. Why is. an ordinary randomized block design inaccurate for comparisons of a large
number of varieties?
2. What principles are involved in incomplete block tests? What is a symmetrical
(or balanced) incomplete block?
3. Explain how to write out the sets for a completely orthogonal i zed 5 by 5 square.
4. What variations in field layout are permissable with a symmetrical incomplete
block test?
5. How does the computation for variety sums of squares differ from that for an,
ordinary randomized block?
6. What is the efficiency factor? Compare the efficiency of a symmetrical incom
plete block test with that for a randomized block trial.
7. What are the limitations in the use of the symmetrical incomplete block design?
8. How would you arrange a variety test so as to be able to fit 47 varieties into
a symmetrical incomplete block test?
Problems
1. It is desired to conduct a symmetrical incomplete block test for lo varieties
of wheat. The form to be used will be m = n'2. A 4 by It orthogonal! zed square
is given below. Write out the sets for the different blocks for each replicate*
111 234 3U2 423
222 143 431 31^
333 [ :12 124 2 lH
444 321 213 132
2. Some uniformity trial data on wheat nursery plots were as follows in grams per
15 foot row (Data from Dr. G. A. Wiebc) :
695 860 960 725 615
735 910 Q75 775 680
645 745 815 700 605
630 $10 730 635 535
680 745 840 730 645
620 730 775 680 610
620 745 660 565 520
560 675 690 635 525
625 706 725 6fc 645
700 765 725 615 640
685 78^ 655 6c6 570
625 556 • 590 590 605
745 790 675 600 625
680 . 670 630 64o 645
655 730 615 650 640
625 700 675 720 695
Use the incomplete block sets written for problem 1 and apply the above yields
to them. Calculate the data for a synmetrical incomplete block design.
CHAPTEB XXI
MECHANICAL PROCEDURE IN FIELD EXPERIMENTATION
I. General Considerat ions
The experimental farm should "be kept neat, clean, and in order at all times. Weeds
should "be hoed from plots anil allays and all trash destroyed. Alleys and roadways
should he hoed or cultivated unless seeded to grass. Straight plot rows add to the
general attractiveness and in some cases to accuracy.
(a) C rop Potation Scheme
Zavitz (1912) states that it is essential to havo a rotation plan for the
entire experimental farm in order to maintain soil fertility. In addition, accurate
maps should he kept for the different fields so that a continuous record exists as
to the crops grown on each field for all past years. A rotation scheme prevents mix
tures in small grain nurseries as well as on other plots since volunteer grain may
contaminate seed plots where the same crop was on the land the previous year. On the
Colorado Station farm it has been found advisable to fallow some of the fields to
equalize the soil moisture duo to the effect of irrigation and for weed control. How
ever, many experiment stations prefer that a bulk crop always precede nursery plots.
At the Nebraska station fallow has failed to equalize soil conditions,
(b ) P reparat ion o f Land for Experimental Crops
All plots for field trials should receive similar treatment except where the
treatment itself is under study. Cultural operations should be at right angles to
the direction of the plot rows so far as practicable. Thome (I909) states that
fertilizers should be applied by machinery rather than by hand methods because of the
more uniform distribution. A twoway plow is useful in seedbed preparation as a
means for the elimination of dead furrows and back furrows in the middle of the ex
perimental area. Seeding machinery used in experimental work must be accurate and,
for that reason, should be calibrated wherever possible. Many machines are unfit for
such work, A drill that fails to drop seed i:.niformly may cause a serious error in
field, plot yields. Moreover, it is very desirable to have a, drill that can bo cleaneC
out readily. Plot rows should be made straight because crooked rows cause irregulari
ty in plot shape.
A  Methods for Planting Experimental Crops
EI ■ S eed P r epar at 1 on
The best sued obtainable should be used in variety trials, i.e., pure as to variety,
free from weed seeds and foreign material, high germination, and uniform in size.
(a) Seed Source
Seed from entirely different sources may entirely upset the small differences
commonly found in yield trials . All seed used in such trials should have been grown,
harvested, and stored under uniform conditions for at least two years, according to
Engledow and Yule (I926) . This is usually impossible. Under those conditions, Par
ker (1931) advises "all that can. be lone is to see that the seed of the several
varieties is approximately of equal germination and. is equally sound and healthy in
other ways. 1 ' Adapted seed is highly desirable for self fertilized crops and often
even more so in cross fertilized crops like corn.
■258
239
(t>) Other Considerations
Unless disease reaction is under study, seeds of cereals should be treated
for control of fungus diseases such as smut. New Improved Ceresan, a dust treatment,
may he used at the rate of one half ounce per bushel for the covered smuts. All
seed should he of the same age when possible. It should he weighed out for the
particular test on the same scales, especially when planted by weight per unit area.
The procedure on many stations is to measure out the seed for both nursery and field
plots. For rodrow or nursery trials the seed is placed in coin envelopes and num
bered to correspond to the plots. When a drill is used, a little more than enough
seed is desirable because the drill itself measures the seed planted.
III. Rate of Seeding
Considerable error may be introduced in some crops through variation in rate of seed
ing .
(a) Small Grains
In small grains the investigator must either plant equal weights or equal
numbers of seeds per unit area. Up until 1910, the "centgener" method was extensive
ly used in small grain nurseries for the determination of yields. The kernels were
spaceplanted 10 inches apart each way in blocks and contained 100 seeds. Aside from
the theoretical objections in genetics, this was an absurd practice from the stand
point of field yields because the seeds were planted approximately lk times as far
apart as ordinarily occur in a drill planted field. In addition, a great amount of
detailed hand labor was required. The method has been discarded in this country in
favor of the rod row. (1) Rod Row Trials: The general procedure in rod rows is to
measure the seed per row. Kiesselbach (1923) summarizes the situation very well.
Fortunately, he states, there may be considerable variation in the rate of seeding
without material effect on the yield per acre. For instance, Turkey wheat planted at
3,1*, 5, 6, and 8 pecks per acre at Nebraska yielded 22.2, 2^.6, 23.7, 2k. h, and 2k. 5
bushels per acre, respectively, for 9 years. Seed of average size, or screened seed,
should be used for machine planters. Measurement of the seed gives results more com
parable with field conditions than where individual seeds are space planted as in
the centgener method. Seed for hand planting should be weighed. (2) The "Checker
board" Trial: The English workers use the "checkerboard" to some extent in their
variety trials . It is essentially a modified centgener plan in which the seeds are
spaced 2x6 inches apart. They admit it differs from field conditions and, for this
reason, use larger "observation" plots to supplement the checkerboard trials. The
checkerboard is precise but requires too much time and labor where many varieties are
under test .
(b) Other Crops
Corn is generally planted by farmers in rows 3.0 to 3.5 feet apart. The usual
rate is three plants per hill for checked corn or with single plants 1^ inches apart
when drilled in the row. Under dryland conditions, the plants are usually drilled 20
to 30 inches between plants in the row. This is the practice in experimental work
except that the seed is often planted at double the required rate, later thinning the
plants to the desired stand. Without this precaution, Kiesselbach (1928) points out,
competition between adjacent rows that differ materially in stand may lead to faulty
results. In sugar beets the seed is generally planted very thick. They are later
thinned tc the desired interval between plants, usually 12 inches. Sugar beets are
ordinarily planted in rows 20 inches apart .
IV. Methods to Plant Field Plots
The ordinary grain drill is often used to plant field plots of 3mall grain and forage
crons .
2^0
(a) C alibration of Grain Dr ills
The necessity for drill calibration was shown "by the work of Bonnet t and
Burkart (1923). The drill may "be jacked up, the seed rate set as desired, and the
wheels turned 30 revolutions at the rate they would turn over in the field. The
amount of grain collected for each drill should "be weighed. A mark should he made
on the wheel to facilitate the coujit . It is only a matter of arithmetic to calculate
the rate that the seed will be planted.
( b ) Use of the Drill
For small grain and forage crops the different replications of the same
variety should be planted before the seed is changed. The plots may be staked out in
advance to facilitate this procedure. The drill should be thoroughly cleaned out
between varieties, possibly by the aid of an air bellows to dislodge seed in the
corners of the drill box. Some drills are made over so that the seed box can be
tipped f orward on hinges to empty. In some experiments where two kinds of seed are
planted in a plot, one crop may be drilled in one direction and the other at right
angles to it, e.g., nurse crop studies in alfalfa.
V. Meth ods to Plant Small Grain Nursery Bows
Small grain nursery plots involve hand methods after the seedbed has been prepared.
Eod rows 12 inches apart are generally used. At some experiment stations 18foot
rows are planted, being trimmed down at harvest time to 16 feet for wheat, 20 feet
for barley, and 15 feet for oats. This enahles the investigator to convert the
yields in grams per plot into bushels per acre by the use of a simple factor. The
rod rows may be made by the use of a sled marker with the runners spaced at the pro
per intervale, the ideal type being horse drawn. The rows are then opened with a
wheel hoe for hand planting. Another method is to use a sugar beet cultivator with
bull tongs spaced at the proper intervale. This has proved to be very satisfactory
at the Colorado station. A 12inch furrow drill is used to mark out the rows on the
Akron Station. The seed, previously weighed out, is sometimes handplanted (scattered)
in the row. A Columbia or planet Jr. planter is used in many cases to plant wheat.
Modification of the Columbia drill for planting oats and barley has been suggested
by Woodward and Tinge.y (1953) as well as by Jodon (1932). A rapid method for plant
ing is by use of the spout drill. This is very satisfactory for genetic material
where yield is not a factor. The grain is poured through the spout, all seed in the
packet being planted in the row length. After a little experience the seed can be
planted very unif orm."iy . One man pushes the drill while another drops the seed. The
spoutdrill may be used for space pi anting after a little practice. One station that
uses 3row plots for nursery studies has a horsedrawn planter. A convenient method
for space planting small grains at definite intervals, for example two inches, is
to take a 6inch board and bore holes at the proper intervals. The seeds are dibbled
in these holes .
VI . Met hods to Pl ant ."Row Crops
Corn will be taken as an example of a row crop. Generally a horsu or hand drawn
marker is used to mark the distances between rows. When the corn is to be •check
planted in hills the plots are cross marked to give a set of squares, the intersec
tions designating the hill locations. Hills are generally spaced y.O or 3*5 feet
apart in all directions. Suitable alleys should be left between blocks to facilitate
cultural and harvest operations. The stakes are distributed along one end of the
plots. The seed sacks or envelopes, with the variety number on them, are distributed
to correspond with the stakes. The numbers should be checked against the planting
plan to avoid mistakes. The seed sacks may be re distributee for each replicate.
Corn is generally planted with a hand planter in yield trials. One of the most
2hl
satisfactory planters is a madeover potato planter.* It is constructed to have a
long, fulllength tin sleeve into which the proper number of kernels is dropped into
the shoe. A nail sack is convenient for carrying the seed. For planting six kernels
per hill, in order to thin to three plants later, it is convenient to plant three
kernels each in two jabs about one inch apart. This facilitates thinning.
B — Field Observations and Care
VII. Value of Field Observations
Intimate knowledge of experimental plots is extremely desirable. In fact, observa
tions during the growing period of crop may be as valuable as the yield data. Dif
ferences due to disease, irregular loss of plants, etc., may account for the varia*.
tion in yield. Plot observations should be made at regular intervals. Notes should
be entered in the field book at once while clear and vivid in the mind. Word descrip
tions should be clear and precise, being made as comparisons in terms of the check
when possible. Sometimes sketches, diagrams or photograx^hs are a better method of
expression than word descriptions.
VIII. Measurement of Plant Character s
Field counts or measurements on certain plant characters given in numbers or cate
gories make excellent comparative field records. Formal "score cards" are apt to
make observations perfunctory. Hence, records should depend upon the particular crop
and the needs that may arise. Some of the more important characteristics usually
recorded are as follows: date emerged, stand, winter survival, date ripe, plant
height, lodging, barren stalks (in com), disease infection, etc. Some of these may
be taken in quantitative measures while categories are required in other cases. When
actual counts are out of the question, a scale of metrics may be employed to convey the
relative intensity of attack of a disease or insect pest. The numbers 1,2,3, and k
may be used to represent, respectively, a slight, moderate, bad, or very severe
attack of rust, mildew, etc. A scale of 1 to k is generally adequate for categorical
data. Further subdivision merely leads to confusion. A very good rust scale is
available in the agronomic field book used by the Division of Cereal Crops and
Diseases, U.S.D.A. Yates ( 193*0 reports a bias between different observers when a
large number of counts were made on wheat culms. The bias differed from observer to
observer and from sample to sample. The same individual should make all counts or
at least all counts on a single replication in order to avoid this form of systematic
error .
IX. Stand Counts and Estimates
In certain crops stand counts are valuable, but this depends largely upon the experi
ment. In forage experiments the counts are often made by the use of square yard or
meter quadrats. These may be permanent quadrats in perennial crop studies. In the
case of winter or spring survival counts in winter wheat, the stand percentage is
usually estimated except in special tests. One person should make all the estimates
due to the large personal error invariably introduced when more than one person makes
them. Estimation in categories such as good, fair, and poor stands may be satis
factory. In plant survival studies, as in winter wheat, a more precise method would
be to space plant the seed in rod rows at 2 inch intervals. However spaced plants
have been observed to kill worse than seeded material. Such tests are valueless for
yield.
*Note': The type used at the Nebraska, Minnesota, and Colorado stations is the "Acme
Segment" potato planter manufactured by the Potato Implement Co., Traverse City,
Mich. It can be slightly modified to make an excellent planter.
2^2
X. Date Headed
There is considerable variation among workers as to the date when a crop should he
considered in head. In small grains, date in head is usually a more reliable index
of earliness or lateness than date ripe. This is particularly true under dryland
conditions where winds may prematurely dry up a variety rather than to allow it to
ripen normally. In wheat, oats, and "barley, some investigators take notes on first
heading, i.e., when 10 per cent of the heads are out of the hoot. A plot is consider
ed fully headed out "by some workers when 75 P®r cent of the plants in the plot are
in full head. Others use a standard as follows: (1) Oats, when the heads are half
out of the hoot; (2) Barley, when the beards are out of the boot; and (3) Wheat,
when the heads show out of the boot. Date in silk or date in tassel are common .
notes in com, date of silking being regarded as a more reliable index of relative
maturity than date of tasseling. It is usual to determine the silking date and con
vert the data to the number of days from planting to onehalf silking. The plots
should be gone over at intervals of one or two days when date in head and similar
notes are taken because some dates nay have to be moved up and others back.
XI . Per cent Lodge d
Data on the differential lodging of small grains is desirable as a measure of stiff
ness of straw. Sometimes after heavy rains or irrigations the soil may be loosened
so that the entire plant falls over. This is not true lodging. A plant has an
inherently weak straw when it bends or breaks over. It is often difficult to arrive
at inherent differences because of soil heterogeneity and its influences. A variety
should be considered lodged when the straw leans an angle of h*) degrees or more be
cause, for practical purposes, grain lodged to such an extent is difficult to harvest.
The per cent of grain so lodged is usually estimated regardless of the cause. Plow
ever, Straw weakness can be detected before the plants lean to an angle of h^ degrees.
Some investigators make notations as to whether the straw is apparently weak, medium,
or strong, and denote the condition categorically by V, M, or S. Under irrigated
conditions, small grains may bo irrigated heavily after heading to induce lodging.
In corn, the relative resistance to lodging is often reported as the percentage of
plants erect at harvest. The percentages may be computed from counts of the numbers
of plants erect. In the interest of uniformity a. plant should be considered erect
when it has not leaned more than 30 degrees from the vertical and which does not
have the stalk broken below the ear. For those who wish to take more detailed records
on lodged plants, it is suggested that such plants be separated into those lodged
because of weak roots (leaning and down plants), and those lodged because of weak
culms (plants broken below the ear) .
XI1  Plant Height
Two men are required to take plant height notes readily, one to make the measurements
and the other to record the results. In the case of small grains such measurements
are ■ generally made just before harvest. Sometimes one measurement is taken per plot
while, at other times, several plants are measured at random. One measurement per
plot is enough when the heights are uniform. A convenient rule is a 1 x 1inch stick
marked, at oneinch intervals to 60 inches. Height notes in corn are often taken in
the fall, but can be taken almost any time after the plants have tasselled out fully.
It can be accomplished with an ordinary rule about 12 feet in length, 2.5 inches wide,
and marked at 3 inch intervals.
XIII. Sogui ng Pl ots
Small grain plots should be thoroughly rogued for admixtures before harvest. The
plots should be gone over several times, particularly when the plants begin to head
2^3
or ripen. Rogues are most conspicuous at such times. It is difficult to rogue "bar
ley out of oats "because the oat plants are generally taller than barley plants. Care
ful work is required to rogue off varieties and off types within a crop. These can
be detected most readily by observed differences in culm height, date of heading,
color of leaves, date ripe, and whether or not awns are present. It is a safe rule
to pull all plants that fail to conform to the majority of the plants in a plot.
XIV. Date Ripe
The date on which a crop ripens is important, particularly in small grains where
earliness is often a desirable feature. Some of the criteria used are given below.
(a) Wheat
• The grain may "be considered ripe when it is hard in the morning. The straw
color is not always a reliable criterion of ripeness.. Those who use straw color as
a criterion generally consider the grain ripe when the first nodes "below the heads
on the main culms have turned brown.
(b) Other Cro ps
In oats the plot is usually considered ripe after practically all of the
heads have turned yellow. The barley crop is generally considered ripe when all .
green has disappeared from the heads. It is difficult to estimate date ripe on small
grain that is badly rusted or lodged as it tends to ripen unevenly and often prema
turely in the case of rust. In corn, date in silk is usually regarded as a more
reliable index of maturity than ripening data in the fall.
C  Methods of Harves ting Experim ental Crop3
XV. Difficulties in Harvestin g
The time of harvesting crops often presents difficulties. Parker (1931) mentions
that one might question the fairness when an early small grain variety is compared
with a check variety that may ripen 10 days or more later. As a rule, plots are
harvested as the varieties ripen, particularly, where there are wide differences in
time of ripening. In some parts of the country, the investigator may be able tc
wait until the latest varieties are ripe so that the entire field may be harvested
at once. Except for extreme differences in time of ripening, it is usually possible
to allow the early varieties to stand without particular damage to them. It may be
desirable in some instances to carry out two separate trials, grouping the early
varieties in one and the late ones in the other. In the case of root and tuber
crops, all varieties may be left in the ground and harvested at the same time without
serious consequences. 'The problem in corn is rather simple because all varieties are
left in the field after becoming ripe so as to dry out. In forage experiments, in
clement weather may interfere with the curing process and require that the hay be
turned several times. As a result, it may dry out unevenly or the leaves shatter.
A possible error in weight might result.
XVI. Methods of Harvesting Field Plo ts
The use of farm machinery is often anticipated for large field plots.
(a) Small Grain Plots
Small grain field plots should be gone over carefully before harvest to be
certain that there are no errors due to defective drilling, rodent, or other injury
that might 'influence the yields. When small grains are badly lodged, it may bo.
necessary to separate the varieties along the margins and push them over into their
respective plots before harvest. Kiesselbach (1928) uses a binder equipped with an
2hh
engine that operates the working parts . At the end of the plot, the horses are
stopped hut the engine continues to operate and clean cut the hinder. In the ab
sence of the engine it is necessary to crank the platform and elevator canvasses by
hand. The small grain shocks should he placed well within the plot to prevent chance
mixtures with adjacent plots should wind scatter some of the bundles. At some sta
tions the bundles are shocked on alternate ends of the plots. The shocks may he
tied with binder twine to minimize the risk. When birds are numerous, shock covers
should be provided. They can be made by sewing together ordinary burlap feed hags.
(b) Corn Yield Tests
The entire plot can be harvested without appreciable error when the plant
stand is 90 per cent or better. Otherwise, it is advisable to reject at harvest all
hills with less than the normal stand, and calculate yields on a perfectstand basis.
Usually the imperfect stand hills are cut with a corn knife and removed from the plot.
A record is then made of the number of perfect stand bills that remain. Sometimes
counts on barren stalks, 2eared stalks, suckers, smutted plants, and lodged plants
are made at this time. For small yield trials, actual harvesting can be done con
veniently with an applepicking ba.g. For large field plots, Kiesselbach (1928) uses
a wagon with a flat rack with partitions built on it. A partition may be placed
lengthwise, through the center and each divided, for instance, into three partitions
where three center rows are harvested for yield (as in prow plots with border rows
discarded) . This allows a separate compartment for each row. Three men can husk,
one man being on each yield row. The compartments on the other side can ho used for
the next plot on the return. At the end of the field, the corn from each plot is
sacked and tagged. Field weights of ear corn are sometimes taken. The corn sacks
may then he either piled in small piles in a shed until air dry, or they may be tied
up on wires in a drying shed (Colorado method). The latter seems to allow the corn to
dry out mere evenly and more quickly. Some stations now have elaborate drying equip
ment where the entire plot yield, can he dried to a moisturefree basis in a relative
ly short t ime ,
( c ) For age . E xp er iiaen t g
Forage plots for hay are almost always cut with a mower when l/'iOacre in
size or larger. The plots may he trimmed evenly on the ends before the regular
cutting time. The material is then raked and removed. Borders between plots are
generally disregarded for large field plots. It is an advantage to be able to start
on one side of the field and. mow through all the series, thus lessening the number of
turns. A man should follow the mower with a fork to be sure that hay is not carried
through the alley from one plot to the next. After the hay has been dried sufficient
ly, a side de livery rake may be used to put it in windows, after which it may be
bunched by a dumprake or by hand. A convenient method to handle; the hay from each
plot is to put it on a wagon or truck on which a sling has been placed. The load is
then weighed, the net weight determined, and the hay unloaded, from the truck by a
cable stacker. A small composite sample may he taken to dry to an airdry basis, or
it may he ground for an immediate moisture determination. For plots l/kOacre in
size or smaller, hay nay he weighed conveniently by a portable platform scales on
which a rack is set. For plots away from the central experiment station, a tripod
and. spring balance affords a good method, to weigh forage plots. A large piece of
canvas is equipped with snaps so that, when the hay is put on it. the sides can be
gathered in and snapped to a ring. It Is then readily hung to the scale.
(d) Sug ar Beet T rials
In sugar beet yield trials, 4 row plots are generally used with the two
center rows harvested for yield. Except In studies on stand, and. certain other in
stances mentioned previously, the plots are commonly harvested on the basis of com
petitive beets, i.e., plants surrounded by plants on all. sides. The tops of the
2k<?
other "beets (noneompetitives) may be chopped off with a hoe "before harvest. The
roots are then pulled with a standard "beet puller. It Is common practice to pull
one replication at a time. The roots without tops are usually weighed in order to
have this component for total plot yield in case this seems to be needed later. The
noncompetitives are then discarded. Two 20root samples may be taken from the non
competitive beets as a sugar sample. The competitive beets are then pulled, topped,
and weighed for each plot. The tare is then subtracted from the field weight of the
roots. When a washer is not available, the tare may be taken in the field. The
sample for tare is first weighed, the roots cleaned with steel brushes, and re
weighed. The difference in weight is the tare. It is believed desirable to calcu
late the tare for each plot separately.
XVII. Harvesting Small Grain Nursery Plots
Competent and continual supervision is necessary in the small grain nursery at har
vest time. Some investigators cleancultivate the alleys between series. Under such
conditions the rod rows are generally trimmed down to remove border effect. In
wheat, for instance, the crop is planted in 18foot rows, one foot being trimmed
from each end of the plot. A string may be stretched across the series at both ends
to designate the discard area to be cut, or a 16foot bamboo pole may be used on
each center row (in 3row plots) so that tho wheat may be cut on both ends of the
pole. Other investigators plant the alleys to some readily distinguishable variety,
thus eliminating the border effect on the ends of the rod rows . The alleys are then
removed before harvest. Hand sickles are used to cut nursery plots. The smooth
edged sickle is most widely used, but a sawtoothed sickle is satisfactory when new.
Where straw yields are taken, grass shears may be used to assure an even cut. Kemp
(1935) has constructed a rodrow harvester of the rotary shear type with which 2 men
may cut 1500 rows per day. The harvested bundles are tied with binder twine,
usually in one place. Strings should be tied with a simple, secure knot. The plot
stake may be tied into the bundle or a tag attached to the string with the plot num
ber on it. The bundles may be tied on a table. Men who tie bundles should tape,
their fingers. Seed plots are often sacked with large paper sacks tied" over the
heads to prevent mixtures. By the aid of a large funnel, 25 pound manila bags are
easily placed over the heads. Sacked bundles should be put under cover as soon as
possible to protect them from rain. Small grain bundles may be either shocked in
the field until they are ready to thresh, or hauled to a shed and hung up to dry.
A drying shed may have wires about four feet apart stretched from one end to the
other at sufficient height so that the bundles can be tied to the wire with heads
down. The bundles should be hung fairly wide apart when they are harvested a little
green. This is particularly true for oats.
XVIII. Harvest of Corn .Breeding Material
Inbred and hybrid strains of corn, which are the result of hand pollination, are
usually harvested after maturity. Individual ears may be collected in the bag over
the ear shoot, and all sacks from the same row tied together with binder twine, the
tying being done with an ordinary sack needle. These sacks are then hung to wires
in the drying shed and allowed to remain there until air dry. This method has. proved,
very satisfactory at the Colorado station.
D — Threshing and Storage
XIX. Methods of Threshing Field and Increase Plots
Small grain field and increase plots are commonly threshed with the standard grain
separator. Kiesselbach (I928) has found it necessary to make miner modifications to
adapt them for this purpose. He lists these changes. as follows:. (1) JRemoval of the
?h6
grain elevator; (2) Elimination of the selffeeder; (3) providing a hinged door at
the foot of the tailings elevator for cleaning out "between plots; (k) replacing the
grain auger with a shakertrough device; (5) removal of the grainsaving auger in
the "blower , where" one exists; (6) equipment with a highpressure air pump and tank
to supply air pressure through a hose' to dislodge grains when the machine is cleaned
out between plots; (7) cutting several holes, with covers, into the sides of the
separator at convenient places to observe the interior and to introduce air pressure
to clean out the separator. Such modifications make ' it easier to clean out the
machine "between plots, thus reducing the chances for mixtures. The chances for mix
ture may he reduced further by threshing all plots of the same variety in succession.
Seed can be saved from the last plot of the variety to be threshed. It is important
to operate the machine uniformly throughout each experiment. The grain per plot is
often weighed on a platform scale at the separator.
XX . Threshing Hursery Plot s
Small grains 3.n yield trials are generally threshed with small nursery threshers,
while genetic material is usually threshed by hand.
(a) Kursery Threshers
Several machines that can be cleaned readily have been devised to thresh
small nursery plots. According to Hayes andGarber (1927) "the chief requisites of
a machine to be used for experimental purposes are as follows: It should be easily
cleanable and, in sc far as possible, tkers should be no ledges or ridges upon which
seeds may lodge. The alternate threshing of different nursery crops is a desirable
procedure. Each of the plots of one strain of wheat may be threshed separately in
rotation and then a strain of oats may be threshed in the same way. At the Minnesota
Experiment Station winter wheat is threshed alternately with barley, and spring wheat
with oats. This plan helps materially to reduce the roguing of accidental mixtures
from the plots." The Cornell machine designed by H. W. Teeter is very satisfactory
for multiplerow plots, while the Kansas machine is widely" Used for rod rows. The
Cornell machine has a shaker, screen, and fan. Its most serious drawback is the ■
difficulty in cleaning it between varieties, however, it can be cleaned more readily
than the Kansas "machine. Recently, Vogel and Johnson (193*0 have developed a new
type of rodrow thresher which is a combination of an overshot cylinder and modified
screenless shaker and fan of an ordinary fanning mill. The grain is further cleaned
by a separate recleaner.. It has been found satisfactory for small grains, peas,
flax, and some grasses. Grain weights are taken after threshing, usually in grams
for rodrow plots.
(b) Hand Threshi ng
In genetic material where it is desired to thresh single plants, threshing
is usually done by hand. A threshing board three feet square is useful for this pur
pose. The frame can be made of 1 x 2 inch material over which a canvas is stretched
tightly. Two blocks, about k x 6 inches in size, are then made and covered on both
sides with corrugated rubber. These work very well for threshing wheat and other
naked grains. Eor barley, it has been found, at the Colorado station that the heads
thresh out better when rolled up in a small canvas cloth (about 9 inches square) and
rubbed. A piece of tin bent to form a fan can be used to blow the chaff out of the
grain ^ striking it on the canvas. Coffman (1935) was able to thresh 100 to V)0
single oat panicles per hour by the use of a light weight closefitting leather glove
on the right hand. The s pikelets aire stripped into a grain jpan where the chaff is
easily blown out.
XXI . Me thods for Shelling C orn
After corn has reached an airdry condition it is ready to shell for final .determina
tions. Genetic material is usually shelled by hand, altho some workers use an
2hj
enclosed single ear sheller. An ordinary corn sbeller is very satisfactory for
yield trials. It should "be enclosed so that the kernels are not scattered when the
ears are shelled. The air dry weight of ear corn should first be taken for the corn
from each plot. A platform scale is often used for such weights. It should be
balanced frequently to keep it in adjustment. The corn is then shelled, the cobs >
being looked over minutely to be sure that all kernels have been recovered. The
shelled corn is then weighed and recorded. A 500gram shrinkage sample is taken at
the Nebraska station and ovendried to a constant weight. The yield of moisturefree
corn is calculated from the percentage of oven dry corn in the shrinkage sample. At
the Colorado station, bushel weight is taken with the standard bushel weight tester,
since bushel weight has been found to be an index of maturity. Moisture determina
tions are made with the TagEcppenstall moisture meter, one sample per plot yield.
XXII. Storage of Seed of Experimental Crops
There are probably as many methods for seed storage of experimental crops as there are
experiment stations. The first requisite is a place safe from mice and insects.
Cabinets with metal drawers probably afford the best storage. It is usually neces
sary to fumigate once or twice per year where grain weevils and other insects are
troublesome. For small seed lots a crystalline compound known as "Antimot" will
effectively control insects. Small grain seed is usually kept in cloth bags, es
pecially seed saved from rodrow tests. Genetic material is commonly stored in coin
envelopes. Seed corn for variety or yield tests may be stored in large bins. Gene
tic and breeding material may be kept either in cloth bags or in envelopes. At the
Nebraska station inbred and hybrid seed corn supplies are kept in large clip envelopes
(6x9 inches in size). These are filed in drawers in serial order. A similar plan
:ls followed at Minnesota.
Referenc es
1. Bonnett, R. K., and Burkett, F. L. Rate of Seeding  A factor in Variety Tests.
Jour. Am. Soc. Agron., 15:l6l171. 1923.
2. Coffman, F. A. A Simple Method of Threshing Single Oat Panicles. Jour. Am. Soc.
Agron., 27:1+98. 1935
3. Engledow. F. L., and Yule, G. U. The Principles and Practices of Yield Trials.
Empire Cotton Growing Corp. 1926. '
it. Hayes, H. K., and Garber, R.J. Breeding Crop Plants, McGrawHill, pp. 132100.
1927.
5. Jodon, N. E. Modifications in the Columbia Drill for Seeding Oats and Barley.
Jour. Am. Soc. Agron., 2l+:328. 1932 .
6. Kemp, H. J. Mechanical Aids to Crop Experiments. Sci. Agr., 15:^88506. 1935.
7. Kiesselbach, T. A. The Mechanical Procedure of Field Experimentation. Jour. Am.
Soc. Agron., 20:^33^+2. 1928.
.8. Love, H. H., and Craig, V. T. Methods used end Results Obtained in Cereal Inves
tigations at the Cornell Station,. Jour. Am. Soc. Agron., 10:1^5137 • 1913.
9. Parker, W. H. The Methods Employed in Variety Tests by the National Institute '
of Agricultural Botany. Jour. Natl. Inst. Agr., Bot., Vol. 3? No. 1. 1931*
10. Standards for the Conduct and Interpretation of Field and Lysimeter Experiments.
Jour. Am. Soc. Agron., 25:803828. 1933.
11. Thome, C. E. Essentials of Successful Field Experimentation. Ohio Agr. Exp.
Sta. Cir. 96. 1909.
£40
12. Vogel, 0. A., and Johnson, A. J. A New Type of Nursery Thresher. Jour. Am. Soc.
Agron., 26:629~6>'). 1934.
15. Wishart, J., and Sanders, H. G. Principles and Practice of Field Experimentation.
Emp. Cotton Growing Corp. pp. 7075* an <i 35100. 1935.
I 1 !. Woodward, E. W., and Tingey, D. C. Improved Modification in the Columbia Drill.
Jour. Am. Soc. Agron., 23:231. 1933.
13. Yates, F.., and Watson, P. J. Observer's Bias in Sampling Observations on Wheat .
Emp . Jour . Exp . Agr . , 2:1 7 i !177 . 193*4. .
16. Zavitz, C. A. Care and. Management of Land used for Experiments with Farm Crops.
Proc. Am. Soc. Agron., 4:122126. 1912 .
Questions for Discus a Ion
1. What precautions are necessary in a crop rotation scheme for experimental crops?
2. How should the seedbed he prepared for experimental crops?
3. Under what conditions should experimental seeds be treated for disease?
4. What is the centgouor method? Checkerboard method? Rod row method?
5. How is corn generally planted for experimental purposes? Sugar beets?
6. Hew would you calibrate a drill?
7. Explain how you would layout, mark, and plant a wheat nursery.. Give all dimen
sions and processes.
8. Why are field observations important? What plant measurements and notes are
generally taken on small grains?
9. What different methods can he used for making stand counts?
10. At what time would you consider wheat, oats, and barley in Head. ? Pipe?
11. How would you take lodging notes in small grains? Corn?
12. What precautions or advice should be given to your assistants when rogulng plots?
13. How would you harvest small grains in a test where the varieties differed widely
in date of ripening? Why?
14. How are large field plots of small grains generally harvested? Corn yield tests?
13. Give the detailed steps for harvesting sugar beet plots for yield.
16. Describe a method for harvesting forage viola tests.
17. Explain in detail how you would harvest snail grain nursery plots.
18. What modifications on an ordinary grain separator are necessary to adapt it for
threshing field plots to prevent mixtures?
19. What are the requisites for a small grain nursery thresher?
20. How would you hand thresh barley heads? Wheat heads? Oat panicles?
Problems
1. It is desired to plant wheat in rod row trials at the rate of 9C lbs. per acre,
the rate used by farmers in the vicinity. The nursery rows are i8 feet long and
12 inches apart. Calculate the amount of seed to weigh out in grams for each row.
2. Suppose the yield from a 1 6foot rod row of wheat is 2op grams. Calculate the
yield per acre.
3. The weight of shelled corn harvested is 23 lbs. on a plot 20 hills long. (a)
When the hills are 36 x 36 inches, calculate the yield per acre for air dry
shelled corn, (b) Calculate the yields per acre on the basis of corn with 13J !
per cent moisture when the original shelled corn contained 132 per cent moisture.
4. Make up a table of factors for the conversion of pounds shelled corn per plot to
bushels of shelled corn per acre when 10 to 20 hills are Harvested. Suppose the
hills to he spaced 36 x 36 inches.
FIELD PLOT TECHNIQUE
APPENDIX
251
Table 1.  Area Under the Normal Curve V
t
A
t
A
t
A
t
A
.00
.50000
.ko
.65542 .
.80
.78815
1.20
.88493
.01
.50399
.41
.65910
.81
.79103
1.21
.88686
.02
.50798
M
.66276
.82
.79389
1.22
.88877
.03
.51197
M
.666^0
.83
.79673
1.23
.89065*
.Ok
.51595+
.44
.67003
.84
.79955
1.24
.89251
• 05
.5199^
A5
.67365
.85
.80234
1.25
.89435
.06
.52392
.k6
.6772!+
.86
.80511
1.26
.89617
.07
.52790
M
.68082
.87
.80785+
1.27
.89796
.08
.53188
.48
.681+39
.88
.81057
1.28
.89973
.09
.53586
M
.68793
.39
.31327
1.29
.90148
.10
.53985
.50
.691I+6
.90
.81 59^
1.30
.90320
.11
.5J+380
.51
.69497 .
.91
.81859
1.31
. 90490
.12
.5V776
.52
.698I+7
.92
.82121
1.32
.90658
.13
.55172
.53
.70194
• 93
.82381
1.33
.90824
.14
.55567
.5^
.70540
.94
.82639
1.34
.90938
.15
.55962
.55
.70884
.95
.82894
1.35
.91149
.16
.56356
.56
.71226
. .96
.83147
I.36
.91309
.17
.56750
.57
.71566
.97
.83398
1.37
.91466
.18
.571^2
.58
.71901+
.98
.83646
1.38
.91621
.19
•57535
• 59
.7221+1
• 99
.83891
1.39
. 91774
.20
.57926
.60
.72575
1.00
.84135
1.40
.91924
.21
.58317
.61
.72Q07
1.01
.84375+
1.41
.92073
".22
.58706
.62
.73237
1.02
.84614
1.42
.92220
.23
.59095+
.63
.73565+
1.03
.34850
1.43
.92364
.2k
.59484
.61+
.73891
1.04
.35083
1.44
.92507
.25
.59871
.65
.71+215+
1.05
.8531^
1.45
.92647
.26
.60257
.66
•7*537
1.06
.85543
1.46
.92786
.27
.60642
.67
.7^857
1.07
.85769
1.47
.92922
.28
.61026
.68
.75175
1.08
.35993
1.48
.93056
.29
.6l409
.69
.75^90
1.09
.86214
1.49
.93139
.50
.61791
.70
.75804
1.10
.86433
1.50
.93319
• 31
.62172
•71
.76115
1.11
.86650
1.51
.93443
.32
.62552
.72
.76424
1.12
.86864
1 "^2
•93575+
• 33
.62930
.73
.76731
1.13
.87076
1.53
.93699
.3*
.63307
.7*
.77035+
lJ.ll
.87286
1.54
.93322
• 35
.63683
.75
•77337
1.15
.87493
1.55
• 939^3
.36
.614058
.76
.77637
1.16
.87698
1.56
. 94062
•37
.64431
.77
.77935+
1.17
.87900
1.57
• 9+179
.38
.64803
.78 .
.78231
1.18
.88100
I.58
.94295
.39
.65173
•79
.78524
1.19
.88298
1.59
. 944o8
yd
x. i
■1.60"
1.61
1.62
1 .by
1.64
1.65
1.66
1.6?
1.68
1.60
1.70
1.71
1.72
1.73
1
,75
1
■ 7°
1
.77
1
78
1
1
.80
1
.81
1
.82
1
.83
1
.84
1
.85
1
.86
1
.87
1
.88
1
.89
1
.QO
1
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1
.92
1
• 93
1
• 94
1
.95
1
.96
1
.97
■1
.98
1
.99
A
.94520
.9^630
.9H738
.94843
Ta"ble 1.  Area Under the No rmal C urve Sy ( Cont.)
t" " ~
2 .kO "
t
.95053
. 95154
.95254
.95352
.95449
.95544
.95637
.95728
.95819
.95907
.95994
.96164
.96246
.96327
.96407
. 964854
,96562
.96638
.96712
; ; 0/
PJl
,96856
.96926
.96995.
.97062
.97123
.97193
.97257
.97320
rvvO1
T'\?{]_
,97441
,97500
.97558
.976I5
.97071
p
00
2
01
2
02
2
03
2
04
2
05
2
06
2
07
n
Od
O
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O
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2
.11
2
.12
^
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2
.16
2
.17
.18
2
. 19
2
.20
2
.21
2
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O
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.2.4
2
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ofi
2
.27
Q
pp.
*—
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2
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2
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2
.32
2
•53
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2
• 35
2
.36
2
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2
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P
. 7 SQ
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■9?T
.97831
.97933
, 97982
. 98030
. 98077
, 98124
,98169
,98214
,98257
. 983OO
.98341
. 98382
. 98422
. 98461
.98500
.9855 7
.98574
.98610
.98645
. 98679
.98713
. 98746
y> i 1 '
08809
,98840
, 98870
. 98899
) 00
o
98956
98985
99010
99036
99061
,99086
,09111
90134
, 99153
2.41
2.42
2.43
2.44
2.45
2 .46
2.47
2.48
2.49
2.50
2.51
2.52
2.53
2.54
2 . o5
2.56
2.57
2.58
d • }y
2.60
2.61
2.62
2.63
2.64
^.05
2.66
2.67
2.63
2.60
2.70
2.71
2.72
2.7'')
2.74
2.7:;
2.70
2.77
2.78
2.79
A
.99180
.99202
.99224
.99245»,
. 99266
.99286
.99505
,00324
.09343
.99361
.99379
.99396
.99413
.99430
. 99446
.90461
.99477
.99492
.09506
.99520
,99334
.99547
.99560
■99373
. 99386
rMiciQ
,99398
.99609
.99621
,99632
3
.9 C
,99653
99664
, 99674
.99683
.99693
. 99702
,99711
.99720
,99728
,90737
t
A
2.30
.997H5
2.81
•99732
2.82
.9976O
2.83
.99767
2.34
„99774
2.35
.90731
2.86
.99738
2.87
% q07Cv",.
.99801
d . 09
.99307
2.00
.99813
2.91
.90619
2 . 92
.99825+
2.93
. 99o3,„
2.94
.99830
2.95
.99841
2 .06
.99846
O Qf7
,99851
2.93
.99856
2.99
.S700.L
3.00
. 098G04
3.01
. 99869
3,02
.09674
3.03
.93873
3.04
■.99882
3 . 05
.90886
7 >.o6
. 99889
3.07
.99893
3.03
.99897
3.09
.99900
310
.99903
3.ll
.99907
3.12
.99910
3.13
.99913
3, 14
.99916
3.13
. '/99 lo
3.16
.99921
3.17
on.' .0)1
5.13
.99O26
3.10
>53
Table 1.  Area Under the Normal Curve ^(Cont . )
A
A
A
A
3.20
.99951
340
.99966
3.60
.99934
3.3o
.99993
3.21
.99931+
3JM
.99967
3.61
.99985
3.81
.99995
3.22
.99956
3.42
.99969
3.62
.99985+
3.32
.99993
5.23
.99938
3A3
.99970
5.63
.99936
3.83
. 99994
3.24
.99940
3.44
.99971
3.64
.99936
3.84'
.99994
5.25
.99942
3^5
.99972
5.65
.99987
3.35
. 99994
5.26
.99944
3.46
.99975
3.66
.99987
3.36
.99994
3.27
.99946
3.47
.99974
5.67
.99938
3.37
•99995
3.28
.999^3
3.48
.99975
3.68
.99988
3.38
.99995
3.29
.99950
3.49
.99976
3.69
.99939
3.89
.99995+
5.50
.99952
3.50
.99977
370
.99989
3.90
•99995*
5.51
.99953
3.51
•99973
5.71
.99990
3.91 .
•99995*
3.32
•99955+
3.52
.99978
3.72
.99990
3.92
.99996
5.53
.99957
3.55
.99979
573
.99990
3.93
.99996
3.54
•99953
3.54
.99980
5.74
•99991
3.94
.99996
3.35
.99960
355
.99961
3.75
.99991
395
.99996
3.36
.99961
5.56
.99982
3.76
.99992
3.96
.99996
5.57
.99962
5.57
.99982
3.77
.99992
397
. 99996
3.38
.99964
5.58
.99983
3.78
.99992
3.93
•99997
5.59
.99965+
559
•99984
3.79
.99993
399
4.00
4.50
. 99997
. ,99997
•99999
^Table I wae taken from "Tables" by L. R. Salvosa, published in "Annals of Mathe
matical Statistics", May 1930.
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3 TV. Neparian
or Hyperbolic ]
jogarithms 1
1 2 3
4 5 6
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12 3 4 5
6789
1.0
0.0000
0100 0198 0296
0392 0488 0583
0677 0770 0862
10 19 29 33 48
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0.2624
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1906 1989 2070
2700 2776 2852
1310 1398 1484
2151 2231 2311
2927 3001 3075
1570 1655 1740
2390 2469 2546
3148 3221 3293
9 17 26 35 44
8 16 24 32 4o
7 15 22 30 37
52 61 70 78
48 56 64 72
44 52 59 67
1.4
1.5
1.6
0.3365
0.4055
0.4700
3436 3507 3577
4121 4l87 4253
4762 4824 4886
3646 3716 3784
4318 4383 4447
4947 5008 5068
3853 3920 3983
4511 4574 4637
5128 5188 5247
7 14 21 28 35
6 13 19 26 32
6 12 18 24 30
4l 48 55 62
59 45 52 58
36 42 48 55
1.7
1.8
1.9
0.5306
0.5878
0.6419
5365 5423 5481
5933 5988 6043
6471 6523 6575
5539 5596 5653
6098 6152 6206
6627 6678 6729
5710 5766 5822
6259 6313 6366
6780 6831 6881
6 11 17 23 29
5 11 16 22 27
5 10 15 20 26
34 4o 46 51
32 38 43 49
31 36 4l 46
2.0
0.6931
6981 7031 7080
7129 7178 7227
7275 7324 7372
5 10 15 20 24
29 34 39 44
2.1
2.2
23
0.7419
0.7885
0.8329
7467 7514 7561
7930 7975 8020
8372 84l6 8459
7608 7655 7701
8065 8109 8154
8502 8544 3587
77^7 7793 7839
8198 8242 8286
8629 8671 3713
5 9 14 19 23
4 9 13 1.3 22
4 9 13 17 21
28 33 37 42
27 31 36 4o
26 30 34 38
2.4
2.5
2.6
0.8755
O.9163
0.9555
8796 8838 8879
9203 9243 9282
9594 9632 9670
8920 8961 9002
9322 9361 9400
9703 9746 9783
9042 9083 9123
9^39 9478 9517
9821 9858 9895
4 8 12 16 20
4 8 12 16 20
4 8 11 15 19
24 29 33 37
24 27 31 35
23 26 30 34
2.7
2.8
2.9
0.9933
1.0296
1.06^7
9969 0006 0043
0332 0367 0403
0682 0716 0750
0080 0116 0152
0438 0473 0503
0784 0818 0852
0188 0225 0260
0543 0578 0613
0886 0919 0953
4 7 11 15 13
4 7 11 14 18
3 7 10 14 17
22 25 29 33
21 25 28 32
20 24 27 31
3.0
1.0986
1019 1053 1036
1119 1151 1184
1217 1249 1282
3 7 10 13 16
20 23 •dS 30
31
32
3.3
1.1314
1.1632
1.1939
1346 1378 1410
1663 1694 1725
1969 2000 2030
1442 1474 1506
1756 1787 1817
2060 2090 2119
1537 1569 1600
1848 1878 1909
2149 2179 2208
3 6 10 13 16
3 6 9 12 15
3 6 9 12 15
19 22 25 29
18 21 25 28
18 21 24 27
3A
35
36
1.2238
I.2528
1.2809
2267 2296 2326
2556 2585 2613
2837 2865 2892
2355 2384 2413
264l 2669 2698
2920 2947 2975
2442 2470 2499
2726 2654 2782
3002 3029 3056
3 6 9 12 15
3 6 8 11 14
3 5 8 11 14
17 20 23 26
17 20 22 25
16 19 22 25
37
33
39
I.3083
1.3350
I.3610
3110 3137 3164
3376 3403 3429
3635 3661 3686
3191 3218 3244
3455 3481 3507
3712 3737 3762
3271 3297 3324
3533 3553 3584
3788 3813 3833
3 5 8 11 13
3 5 8 10 13
35 3 10 13
16 19 21 24
16 18 21 23
15 18 20 23
4.o
1.3863
3888 3913 3938
3962 3987 4012
4036 4o6l 4o85
25 7 10 12
15 17 20 22
k.i
k.2
4.3
1.4110
1.4351
1.4586
4134 4159 *H8^
4375 4398 4422
4609 ^633 4656
4207 4231 4255
4446 4469 4493
4679 ^702 4725
4279 4303 4327
4516 454o 4563
4748 4770 4793
2 5 '7 10 12
2 5 7 9 12
2 5 7 9 12
14 17 19 22
14 16 19 '21
14 16 18 21
4.4
4.6
1.4816
1.5041
I.52&I
4839 ^861 4884
5063 5085 5107
5282 5304 5326
4907 4929 4951
5129 5151 5173
5347 5369 5390
i).'974 4996 5019
5195 5217 5239
5412 5433 5^54
2 57 9 11
2 4 7 9 11
2 4 6 9 11
14 16 18 20
13 15 18*20
13 15 17 19
260
4.8
5.0
5.1
52
5. 3
5.^
55
5.6
57
5.8
59
6.0
6,1
6.2
6.5
6.14
6.5
6.6
6.7
6.8
6.9
o
l . 5476
1.5686
l . 5892
. 609)+
1.6292
1.6487
I.6677
1.6864
1.70^7
1.7228
1.7405
1.7579
1.7750
1.7918
I.8085
1.8245
1.8405
1
Table IV. Neparian or Hyp erb olic Logari thms! (Cont.)
_
3
5497 5518 5559
5707 5728 5748
5915 5935 5953
6ll4 6134 6154
6312 6332 6351
6506 6525 6544
6696 6715 6734
6882 6901 6919
7066 7084 7102
7246 7263 7281
7422 744o 7457
7596 7613 7630
7766 7785 7800
7934 7951 7967
8099
8262
8421
8116 8132
8278 8294
3437 8433
1.8565 8579
I.8718 I 87o3
1.8871 8886
S594 8610
8749 876^:
8901 8916
1.9021 1 9036
I.9169 9184
1.9315 '9356
7.0
7.1
7.2
7.3
7.4
75
76
77
7.8
7.9
3.0
8.1
8.2
8.3
1 n)
9459
1.9601
1.9741
1.9879
9051 9066
9199 9213
9^44 9339
9473 9488 950.
9615
9755
98Q2
9629 96!: 3
9769 9782
9906 9920
2.0015
2.0149
2.0281
2.0412
2.0541
2.0669
2.0794
2.0919
2.1041
2. ].l63
0028
0162
0295
0042 0055
OI76 OI89
0308 0321
0425
0554
0681
0438 0451
0567 0580
0694 0707
4
5560 5581 5602
5769 5790 5810
5974 5994 5oi4
6174 6194 6214
6371 6390 6409
6563 6582 6601
6752 6771 6790
6938 6956 6974
7120
7133
7299 7317
7156
733 1 )
7475 7492 7509
7647 7664 7681
7817 7834 7831
7984 3001 8017
8148 8165
8310 8326
8469 8485
8131
8342
8500
8625 8641
8779 8795
8931 3946
8656
8310
8961
9081 9095
9228 9242
9373 9387
9110
9237
q4o2
9516 9530 9544
9657 9671
9796 9810
9933 99^7
9685
9824
9961
0069 0082
0202 0215
0334 0347
0096
0229
0807 0819 0832
0931
1054
1175
0943 0956
1066 1078
1187 1199
0464 0477
0592 0605
0719 0732
C490
0618
0744
0844 0857 0869
0968 0980
1090 1102
1211 1223
0992
1114
1235
7
8 9
5623 5644 5665
5831 5851 5872
6034 6054 6074
6233 6253 6273
6429 6448 6467
6620 6639 6658
6808 6827 6845
6993 7011 7029
7174 7192 7210
7352 7370 7337
12 3
2 4 6
2 4 6
I 4 6
4 6
2 4 6
2 4 6
2 4 6
2 4
2 4
2 4
7527 7544 7561 I 2 3
7699 7716 7733 2 3
7867 7884 7901 2 3
3034 8050 8066
8197 3213
3358
8516
3374
8532
8229 ! 2
8390 J 2
8347 2
1
3__5_
3 5
3 5
3 5
3 11 13
3 10 12
8 10 12
8 10 12
8 10 12
8 10 11
7 9 11
7 9 11
7 9 11
7 9 11
7 9 10
7 9 10
7 3 10
7 8 10
7
A
9
15 17 19
14 16 19
14 16 18
14 16 18
14 16 1.3
13 15 17
13 15 17
13 15 16
13 14 16
12 14 16
12 14 16
12 14 13
12 13 15
12 13 15
6 3 10 ill 13 15
6 8 10 li 13 14
6 8 9 11 13 14
8672
3825
8976
9125
9272
9416
8687
8840
8991
9l4o
9286
9430
9539 9373
8703 I 2
8856 I 2
9006 J 2
i ™
9135 I 1
9301 J 1
9449 j 1
9587 i 1
i b
6
! 6
r
3 k
3 .4
9699
9838
9974
0109
0242
9713
985I
9988
9727
9865
0001
3 4
a h
o
i 6
!6
8
3
3
11 12 14
11 12 14
11 12 14
9 10 12 13
9 10 12 13
9 10 12 13
9 1 10 11 13
7 8
7 8
7 8
10 11 13
10 11
10 11
12
12
012.2
0255
0136 ! 1
0268 ! 1
4 I.5
>373 0386 0399 \ l
0503
0631
0757
0516
0643
0769
0528 i 1
0656 I 1
0782 I 1
08S2 0894 0906 1 3
'7 8
7 8
l 5 6 8
5 6 3
568
5
o
8
1005 1017 1029
112b 1133 1150 .
1247 1258 1270 j 1
2 4 15 6 7
2 4 j 5 6 7
2 4 ! 3 6 7
9 11 12
9 11 12
9 10 12
9 10 12
9 10 12
9 10 11
9 10 11
9 10 11
9 10 11
8 10 11
261
•
Table 17.
Neparian or Hyperbolic Logarit
bms 1 (Cont.)
12 3
1+5 6
7 8 9
12 3
1+
5 6 7
8 9
8.1+
8.5
S.6
2.1282
2.11+01
2.1518
129"+ 1306 1313
11+12 ll+2l! l'!3o
1529 15^1 1552
1330 13^2 1353 I1365 1377 1369
11+1+8 11+59 1^71 j 11+33 ll+9'+ 1506
1561+ 1576 1587 3.599 1610 1622
1 2 k
12 4
12 3
6 7
6 7
6 7
8
6
O
O
10 11
9 11
9 10
8.7
3.8
3.9
2.1633
2.171+8
2.1861
161+5 I656 1668
1759 1770 1782
1872 1833 189!+
1679 1691 1702
1793 1801+ 1815
1905 1917 1923
1713 1725 1736
1827 1838 I3lf9
1939 1950 1961.
1 2 3
12 3
1 2 .3
5
5
k
6 7
6 7
6 7
8
8
8
9 10
9 10
10
9.0
2.1972
I983 199^ 2006
2017 2028 2059 1 2050 2061 2072.
1 2 3
1+
6 7
p.
9 10
Q.l
9.2
9.3
2.2083
2 .2192
2.2300
209I+ 2105 2116
2203 2211+ 2225
2311 2322 2332
t
2127 2138 23J+3J2159 2170 2181
2235 221+6 225712268 2279 22o9
23l(3 235I+ 236412375 2386 2396
12 3
1.2*3
12 3
1+
1+
1+
i
5 7 1 8
5 6 j 8
5 6  7
9 10
9 in
9 10
9.*
95
9.6
2. 21+07
2.2513
2.2618
21+18 21+28 2I+39
2523 253I+ 25V+
2628 2638 261+9
21+50 21+60 2 1+71 2l+3l 21+92 2502
2555 2565 2?7S t ?536 2597 2607
2659 2670 2t.80 i 269c 2701 2711
12 3
12 3
1 2 3
1+
1+
1+
5 6 i 7
'5  7
"5 67
10
O n
y
3 9
9.7
9.8
9.9
2.2721
2.2821+
2.2925
2732 27^2 2752 2762 2773 27&3J2795 2803 231 1+
283I+ 281+1: 285I+ 2865 2875 2385 2895 2905 2915
2935 29I+6 2956:2966 2976 2986 2996 5006 3016
■ 1 . . j
1 2 3
1 2 7 
12 3
1+
k
h
5 6 ! 7
5 6 i 7
5 67
8
8 9
j
Table of Neperian Logarithms of 10
+11
n
1
3
k  ■ ,
6'
7
• 8
9
log e 10 n
2.3026
1^.6052
6.9078
1
9.2103 11. 5129
13.3155
16.1181
18.1+207
20.7233
Table of Neperian Logarithms of 10
n
11
1
2
3
k
5
6
7
9
log e 10 n
3.6971+
5.391+8
7.0922
IO.7897
12 .1+871
14 . 181+5
17 .8819
195793
21 ,276".
^"This table is reproduced from "Four Figure Mathematical Tables" by the lace J. T
Bottomley and published by Macmillan and Co., Ltd. ( London) . The consent of the
publishers and representatives of the author have been obtained.
(352039)
262
Table V. Values of percentages transformed into degrees of an angle. Angles of
equal information are given in the "body of the table corresponding to observed per*
contages along the left margin arid top. (hach angle ending in 5 is followed by a 
or a  sign for guidance when the last decimal is dropped) .
Table taken from article by Dr. Chester I. Bliss of the Institute for Plant Protec
tion; Moscow, Russia. Reproduced by permission of the author.
/
0.00
. 01
0.02
. 03
0.Q4
0.03
0.06
0.0T
.00
. 09
0.0
0.57
o.Si
0.99
1.15"
1.23
1.40
1 . 52
1.62
1 . 72
0.1
1.81
1.90
1.99
2.0T
2 . 14
2.22
2 .29
2 . 56
2.43
2.30
0.2
2.56
2.63
2.69
2.75
2.8.1
2.87
2.02
2.98
3.03
3.09
0.3
3.1^
3 . 19
3.24
3.29
3.2*
3.39
3.44
3.49
3.93
3.33
0.4
3.63
3.67
3.72
3.76
3.8c
5.83
3.89
393
3.97
4.01
0.5
it. 05+
k . 09
4,13
4 . 17
4.21
4.25+
4 .20
4 . 35
4.3T
4.40
0.6
4,44
4.48
4 . 52
4 . 55+
4.59
it. 62
4 . 66
4 . 69
! +.73
4 . 76
o.T
4.80
4.83
4.87
4.90
4 . 95
4 . 9T
s no
5.03
5.07
5 . 10
0.8
5.13
5.16
5.20
r~, ^i'/
5 • 2
5.20
5.32
5.33+
3. 53
5.41
0.9
5.44
5 AT
5 . 30
5.33
5 • 36
3.59
1 ,'Vo
5.634.
3 . 71
9^6
O.63
9.81
11 .0°
11.24
11.59
12.52
12.65
12.79
13.31
1394
Ik, 06
15.00
15.12
15.23
lo.ll
1.6.22
le.32
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 5.74 6.02 6.29 6.^ 6.80 7.04 7.27 T.4S 7.71
2 8.15 3.55 8.55 6.72 3.91 9.10 9.28
3 9.98 10.14 10.51 10. 47 10.65 10.78 10.94
4 11,54 11.68 11.85 11.97 12.11 12.25 12.59
5 12.92 13.05+ 13.16 13.51 15. H 1.5.56 13.69
6 14.16 lU.30 14.42 14.54 14.6?+ 14. TT 14.39
T 15.34 15.45+ 15.56 13.68 15. '79 15,89 16.00
8 16.45 16.54 16.64 16.74 16.65 16.95+ 17.05+ 17.16 17.26 17.36
9 17.46 17.56 17.66 17.76 17. 8 C ^ 17.93+ 16.05 18.15 13.24 18.54
10 13. 44 18.55 I8.63 18.72 18.81 16. 91 19.00 19.09 I9.I9 19.28
11 19.37 19.46 19.551 19.64 19.75 19.82 19.91 20.00 20.09 20.13
12 20.27 20.36 20..44 20.53 20,62 20/70 20.79 20.86 20.96 21.05
15 21.13 21.22 21.30 21.39 21.47 21.56 21.64 21.72 21.31 21.89
14 21.97 22.06 22.11 22.22 22.30 22.33 22.46 22.35 22„53 22.71
15 22 .79 22.67 22, 9e 25,03 23.ll 23.19 23.26 23.36 23.42 23.50
lb 23.58 25.60 25.73 23.61 23.89 23.97 24.04 24.12 24.20 dh .27
1.7 24.35+ 24.43 24.50 24.58 24.65+ 24.73 24.80 24.33 24.05 25.05
18 25.10 25.18 23.25+ 25.33 23.40 25.48 25.55 25.62 25. TO 25.77
1.9 25.84 25.92 25.99 26.06 26.15 26.21 26.26 26.55 26.42 26.49
20 26.3c 26.64 26.71 26.78 26.85+ 26.92 26.99 27.O0
21 27.28 27.30 27.42 27,49
22 27,97 28. Oil 28.11 26.18
23 23.66 26.73 28.79 28. 8e
24 29.35 29A0 29.47 29.55
27.56
27.65
2 r " 60
27. 76
28.25
28.32
28.45
23.93
29.00
29.06
29.13
29.60
29.67
29. T5
29.80
27.15
27.20
27.65
28.32
27.00
P. en
29.20
29.87
29.27
29.95
263
Table V. Values of percentages transformed into degrees of an angle. Angles of
equal information are given in the "body of the table corresponding to observed per
centages along the left margin and top. (Continued)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
25
30.00
30.07
30.13
30.20
30.26
30.33
30.4o
30.46
30.53
30.59
26
30.66
30.72
30.79
30.85+
30.92
30.98
31.05
31.11
31.18
31.24
27
31.31
31.37
31.44
31.50
31.56
31.63
31.69
31.76
51.82
31.88
28
31.95
32.01
32.08
32.14
32.20
32.27
32.33
32.39
52.46
32.52
29
32.58
32.65
'32.71
32.77
32.83
32.90
32.96
.33.02
35.09
33.15
30
3321
33.27
33.3^
33.40
33.46
33.52
33.58
33.65
33.71
33.77
31
33.83
33.89
33.96
34.02
34.08
34.14
34.20
34.27
54.55
34.39
32
 3kM
34.51
3*. 57
34.63
34.70
34.76
34.82
34.88*
54.94
35.00
33
■ 35.06
35.12
35.18
35.24
35.30
35.37
35.43
35.49
55.55
35.61
34
35.67
35.73
3579
35.85
35.91
35.97.
36.03
36.09
56.15+
36.21
35
36.27
36.33
36.39
36.45+
36.51
36.57
36.63
36.69
56.75+
36.81
36
36.87
36.93
36.99
37.05
37.11
37.17
37.23
37.29
37.35
37.41
37
57.47
37.52
37.58
37.64
37.70
37.76
37.82
37.88
37.94
38.OO
38
38.06
38.12
38.17
38.23
38.29
38.35*
38.41
38.47
38.55
38.59
39
38.65
38.70
38.76
38.82
38.88
38.94
39.00
39.06
39.11
39.17
4o
39.23
39.29
39.35
39.41
39.47
39.52
39.58
59.64
39.70
39.76
4l
39.82
39.87
39.93
39.99
40.05
40.11
40. 16
40.22
4o.28
40.54
42
4o.4o
40.46
40.51
40.57
40.63
40.69
40.74
4o.8o
40.86
40.92
43
40.98
41.03
41.09
41.15
41.21
41.27
41.32
41.38
41.44
41.50
44
41.55+
4l.6l
41.67
41.73
41.78
41.84
41.90
41.96
42.02
42.07
45
42.15
42.19
42.25
42.30
42.36
42 .42
42.48
42.53
42.59
42.65
46
42.71
42.76
42.82
42.88
42.94
42.99
43.05
43.ll
43.17
45.22
47
43.28
43.34
43.39
43.45+
43.51
43.57
43.62
43.68
43.74
45.80
48
43.85+
43.91
43.97
44.03
44.08
44.14
44.20
44.25+
44.31
44.57
49
44.43
44.48
44.54
44.60
44.66
44.71
44.77
44.83
44.89
44.94
50
45.00
45.06
45.ll
45.17
45.23
45.29
45.34
45.40
4546
45.52
51
45.57
45.63
45.69
45.75
45.80
45.86
45.92
45.97
46.03
46.09
52
46.15
46.20
46.26
46.32
46.38
46.43
46.49
46.55
46.61
46.66
53
46.72
46.78
46.83
46.89
46.95+
47 .01
47.06
47.12
47.18
47.24
54
47.29
47.35+
47.41
47.47
47.52
47.58
47.64
47.70
47.75+
47.81
55
47.87
47.93
47.98
48.04
48.10
48.16
48.22
48.27
48.55
48.39
56
48.45
48.50
48.56
48.62
48.68
48.73
48.79
48.85+ 48.91
48.97
57
49.02
49.08
49.14
49.20
49.26
49.31
'49.37
49.43
49.49
49.54
58
49.60
49.66
49.72
49.78
49.84
49.89
49.95+
50.01
50.07
50.15
59
50.18
50.24'
50.30
50.36
50.42
50.48
50.53
50.59
50.65+
50.71
60
50.77
50.83
50.89
50.94
51.00
51.06
51.12
51.18
51.24
51.50
61
51.35+ 51.41
51.47
51.53
51.59
51.65
51.71
51.77
51.85
51.88
62
51.94
52.00
52.06
52.12
52.18
52.24
52.30
52.56
52.42
52.48
63
52.53
52.59
52.65+
52.71
52.77
52.83
52.89
52.95+
5301
53.07
64
53.13
53.19
53.25
53.31
53.37
53.43
53.49
55.55
53.61
53.67
2€h
Table Y. Values of percentages transformed into degrees of an angle. Angles of
equal information are given in the "body of the table corresponding to observed per
cent ages along the left margin and top. (Continued)
0,0 0.1 0.2 0,5 o.k 0.5 0.6 0.7 0.8 0.9
63 5373 5379 5335 53.91 5397 3+.03 5+.09 54.15+ 5^.21 ^+.27
66 5+33 5^.39 54, 1*5+ 54,51 5 J +57 54.63 54.70 5^.76 54.82 54.88
67 5 ] ^9^ 55.00 53.06 55.12 5513 55. 24 3530 3537 55 A3 53 +9
68 5555+ 55.61 55.67 55.73 55.80 55.86 5592 55.98 56. 04 56.ll
69 56.17 56.23 56.29 56.35+ 56. 1*2 56A3 56.34 56.60 56.66 56.73
70 56.79 56.83+ 56.91 56.98 57. 04 5710 57.17 57.23 57.29 57,35+
71 57 M 57.1*8 37.5^ 5761 57.67 5775 57.30 57.86 57.92 5799
72 58.05+ 53.12 58.18 58.2)4 58.31 58.37 58 ,1*1* 58.30 5Q.56 58.63
73 58.69 58.76 58.82 58.89 58.95+ 59.02 59.08 59.15 5921 5928
74 59*3+ 59 +1 39.47 39. 5 J + 39.60 59.67 59. '~(k 59.80 59.87 3993
75 60.00 60.07 60.13 '60.20 60.27 '60.33 60A0 60A7 60.53 60.60
76 60.67 60.73 60.80 60.87 60.9+ 6.1.00 61.07 61.1)4 61.21 61.27
77 61.3)1 61.1*1 61. 1*8 61.55 61.62 61.68 61.75+61.82 61.89 61.96
78 62.03 62.10 &2.17 62.21* 62.31 62.37 62.1*)+ 62.51 62.58 62.65+
79 62.72 62.80 62.87 62.9)4 63.01 65.08 63.15 63.22 63.29 63.36
80 63.1*1* 63.51 63.38 63.65+ 63.72 63.79 63.37 63.94 6)*. 01 61*. 08
81 64.16 64. 23 6'+. 30 64.38 64.45+ 64.52 64.60 6)+. 67 64.75 64.82
82 64.90 bk. 97 65.05 65.I.2 63.20 65.27 65.35 65.1*2 65.50 65.57
83 65.65 65.73 65,80 65. 83 65.96 66.03 66.11 66.19 66.27 66.3)*
81* 66.42 66.50 66.58 66.66 66.74 66. 81 66.89 66.97 67.05+ 67.13
85 67.21 67.29 67.37 67,45+ 67.5'^ 67.62 67.70 07.78 67.86 67.9+
86 68.03 63.11 68.19 68.28 68.36 68.44 68,53 68,61 68.70 68.78
87 68,87 68.95+ 09.04 69.12 69.2.1 69.30 69.38 69,1*7 69.56 69.64
88 69.73 69.82 69.91 70.00 70.09 70.18 70.27 70.36 70.45 70.54
89 70.63 70.72 70.81 70.91 71.00 71.09 71.19 71.28 71.37 71.47
90 71.53 71.66 71.76 71.85+ 71.95+ 72.03 72.15 72.2)+  72.31* 72.1*1*
91 72.54 72.61* 72.74 72.84 72.95 73.03 75.15+ 73.26 73o6 73,1*6
92 73.57 7368 73.78 73.89 74.OO 74.11 7 J *.2l 74.32 'jkM 7 ] '.55
93 74.66 7 ) *77 7^88 75.00 75 .11 75.23 7535 7:3.1+6 75.53 7370
94. 75.82 75.9)* 76.06 76.19 76.31 76.1*1* 76.56 76.69 76.32 76.95
95 77.08 77.21 77.31* 77,1*8 77.61 77.75+ 77.89 78.03 78.17 73.32
96 73.1*6 78.61 78.76 78.91 79.06 79.22 79.37 7955 79.69 79.36
97 80.02 30.19 80.37 80.5)4 30.72 80.90 81.09 81.23 81J+7 81.67
98 81.87 82.08 82.29 82.51 82.73 82.96 83.20 83.1*5+ 83.71 85.98
265
Table V. Values of percentages transformed into degrees of an angle. Angles of
equal information are given in the "body of the table corresponding to observed per
centages along the left margin and top. (Continued)
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
990
99.1
992
995
994
84.26
84.56
84.87
85.20
85.56
84.29
84.59
84.90
85.24
85.60
84.32
84.62
84.93
85.27
85.63
84.35
84.65
84.97
85.31
85.67
84.38
84.68
85.00
85.34
85.71
84.41
84.71
85.05
85.58
85.75
8U.44
84.74
85.07
85. 41
85.79
84. U7
84.77
85.IO
85.45
85.85
84.50
84.80
85.13
85.48
85.87
84.55
84.84
85.17
35.52
85.91
995
99.6
997
99.8
999
100.0
85.95
86.57
86.86
87.44
88.19
90.00
85.99
86.42
86. 91
87.50
88.28
86.03
86.47
86.97
87.57
88.38
86.07
86.51
87.02
87.64
88.48
86.11
86.56
87.08
87.71
88.60
86.15
86.61
87.15
87.78
88.72
86.20
36.66
87.19
87.86
88.85+
86.24
86.71
87.25+
87.95
89.01
86.28
86.76
87.31
88.01
89.19
86.35
36.81
87.57
38.10
89.43
(50758)
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267
SUBJECT INDEX
Page
Ac cl imat i zat i on ; \ ; ; .,„•. 155
Acid phosphate ;;,.... ; ; ; '33
Agricultural research, support for 2
Agricultural science, development of : ; " 9
Agronomic experiments, criticisms of : 167
general types of .19
Agronomi c re sear ch , re suit s of .: 5
rise of „ :.: :: . 1
status of 1 8
trends in , : . : 6
Analogy „ 2k
Analysis of covariance, general 113
computation of 11^
use of , 119
Analy s i s of variance , general , n 103112
one criterion of classification 103
two or more criteria of classification 108
and agricultural experiments 110
for computation of heterogeneity I3I4
Angular transformation, table of 262
Areas under normal curve, table of „ 251
Arithmetic average or mean :.....:: .. 38
Bacteriology, development of 13
Basic plant sciences, history of '. .:...: " 9lS
Binomial d i s t r ibut ion '. : 70 7^
Binomial distribution, applications of '71
nature of  .; k2
Border effect : 162
Burnt limestone ; : 33
Calciummagnesium ratio ; : :,..'. ; .■ '. 32
Calibration of grain drills : 2k0
Cell in relation to inheritance :.... 12
Chi square (x 2 ) : 7586
Chi square (X 2 ), applied to several genetic families ....'. 78
correction for continuity :.'.:.: 83
distribution of „ : 75
null hypothes is and '., 83
partition into components 80
probability tables for : 76
table of : ; _ :. 258
use in homogeneity test 20o
C oef f i c ient of var iab i 1 ity ^5
Competition,, and other plant errors :...■: .;... :•:•:. :.„.'! 55 166
concept ,..::*. ....' .".;. L...... 156
Complex experiments, application to variety trials •.:.;.............. .".:..■■„'....'....!. 195210
versus simple experiments •....... ,...........':.'.....'..'..: loo
Confounding, in factorial experiments ;. ....;. , ■ 221229
in 2 by 2 by 2 experiment '.' ..:..: 223
partial in 2 by 2 by 2 experiment 226
263
Page
Constants, used to describe distributions „. . kj
of binomial distribution  70
C ont inuous select ion 3 2
Corrections for uneven stands 159
Correlation, nature of : B7
Correlation coefficient ■..■. o9>132
Correlation coefficient, calculate on of 115
for error of a difference $h
significance of • 95
Correlation surface 91
Co variance, analysis of 115
Crop rotation experiments 17°
Crucial tests ... . 22
Cultural experiments , 177
Date headed (small grains) 2*t2
Date ripe (small grains) 2*43
Deep plowing 52
Degrees of freedom 62
Design, "basic principles of X&J
relation of type of experiment to 17 6
Differences in stand in plots , .....158
Discovery, methods of : : 2*4
Duration of tests lo9
Dust mulch 51
Early agronomic experiments o
Efficiency factor 230
Empirical iaethcd ■,,... 19
Error, control of 171
reduction of with uniformity data : 120
sources of : 28
types of . 2b"
Errors, in experimental work ^<o.>
related to plant : 29
related to soil N ■... 3°
Evidence ' 22
Evolution and genetics • ,  11
Experiments., crop rotation 17°
cultural ■ , . 17/
factorial ,...; •. 221
fertilizer , ■ 177
pasture 17"
Experiment stations, establishment of :
funds for , 5
Factorial experiments ■  221
Fallacies in agi'omomy 31
Fertilizer experiments 1 II
Field experiment ,' nature of ^0
sources of variation in , : 2Q
Field observations 241
Field plots : l^ 2
Fit of observed data to normal curve ■ 19
Frequency distribution 37
2 69
Page
Genetics , modern developments in„ „ 13
Generalized standard error methods . 103
Goodness of fit, chi square test for 77
Graphi cal repre s entat i on „ kl
Grouping of data kO
Harvesting experimental plots _ 2kj>
Homogeneity test ....; „ 206
Hybridization of plants ....; 11
Hypothesis, formulation of 21
null : 21
qual i t ies of _ _ ,„ 2 1
Incomplete experimental records : 179
Incomplete "block experiments 230
Independence , test for ., 8l
Inductive method 17
Inter plot compet i t i on _ : 160
Intra plot competition : 157
Laboratory and greenhouse experiments. 20
Land preparation _ „ ,. 1 238
Large sample theory , „ 55
Lat in square „ „ , ,...,., 17^
Laws of inheritance _. 13
Logic in experimentation 1727
Mean of replicated variates ....„ , 39
Means , of 2 independent samples   63
of paired samples _ 65
Measurements _ 37 *2i+l
Measures of central tendency _  ^2
Mechanical procedure in field experimentation _ 2382*48
Mendelian ratios, standard errors of 72
Methods, to plant field plots '.... ~ 239
to plant nursery plots 2^0
to plant row crops  2k0
Missing values .: 179
Moisture content of harvested crop : 155
Neparian logarithms, table of :...: 259
Nitrogen in plants ~ ■  : ». : 10
Normal distr ibut ion : ^2
Null hypothesis „  21,83
Nursery plots _ „ 1... '. 1^2
Outline of experimental tests : , I67
Pasture experiment s 178
Percent lodged ~    • 2^2
Percentage data, class if icat ion of 207
trans format ion of ..... : 207
Personal equation „  _. ^,29
Plant competition  155
Plant height 2^2
Plant individuality ....15^
270
Page
Plant nutrition • 10
Plant pathology lit
Plots, early use of : : * : .,':....';... ll0.
kinds of ; Ik?.
size and shape '. ,. 1^1
Plot arrangements ; ' '.' ', 170
Plot efficiency, calculation of : IV7
Plot replication lk$)
Plot shape, practical considerations in : ,. : 1J+7
relation to accuracy Xk6
Plot size, factors that influence ' ; lUl
for various crops .' ll(5
relation to accuracy ,.., , lk$
Pol s son di st r i "but i on .' ^3>73
I'otometers 51
Preliminary tests : I08
Principle of extremes ' , I67
Probable errors of statistical constants 61.
Probability, and normal curve , 55
determinations with small samples 62
relation of binomial distribution to 70
tables for , : )b
theory of ; .'.....'. "A
Quadrat and other sampling methods l86~19^
Questionnaires and surveys ' : 20
Random numbers 266
Randomized blocks 172
Rate and date tests ! ".....! lol.
Rate of seeding , 239
Regression, linear ■ ■ Qo
computation for grouped data ■ ' 98
significance of 99
Regression coefficients, calculation of ". 115
Regression equation, substitution in ' 118
Replication, history of '. lk$
in experimental work ; 1^9
reduction of error by 150
relation to soil heterogeneity 7 169
Replications, number of , lpl
Research 17
Roguing plots _.....;'. 2^2
Rules for computet icn ; 38
Sampling, economy in ; 188
practices used in 190
Science, scope of 17
among ancients ■ ' : 17
Seed preparation 258
Selection of seed com ,.. J2
Sex in plants ? 11
Sheppard'e correction ' 1+5
Shortcut methods for commutation ho
271
S Ignif i cance , le vel s of _ 5^
of means 63
of correlation coefficients 95
Significant differences 60
Simple vs. complex experiments 168
Small samples, special case of 62
in biological research 62
Soil heterogeneity, amount of _ 135
causes of 136
correct ions for 137
relation to experimental field 137
universal ity of 131
Split plot experiments 211220
Stand counts and estimates 2U1
Standard deviat i on .... ^3
Standard errors, of statistical constants 5760
of Mendel ian ratios 72
Statistical terms 37
Statistical methods, general applicability ^7
St at ist i cs , in experimental work .". 37
as basis for generalization ..._ ■. 5^+
Storage of seed 2^7
Student ' s pairing method , 65
Symmetrical incomplete block experiments 230237
Tables . 251266
Theory of evolut ion 1 1
Threshing, field plots and nursery plots 2^5
Types of frequency distributions ^2
Uneven plant distribution _ 157
Uneven stands, corrections for ,. 159
Uniformity trial data  13 1
Values of n F " and "t " 25^
Variance, analysis of 103
Variations in seasons 30
Variety and similar tests 176
y
_i
UNIVERSITY OF FLORIDA
3 12b2 DMM1015M D
MARS,ON SCENCE UBRARy
Date Due
Due
Returned
Due
AUB 2 519ft
OCT 2 31 9
OR 4WS
JAW *'9
L
24 1978
WOf 1 8 WW
»4
— ~
MAfl 8 PTBw 8 «w
2Q 199
/ DEC n 7
2QM.
LIBMRY ^
Returned